Systems and methods for dual comb spectroscopy

ABSTRACT

A method for adaptive dual frequency-comb spectroscopy includes repeatedly (i) recording a single interferogram with a dual frequency-comb spectrometer, (ii) averaging the single interferogram into an averaged interferogram, and (iii) determining a signal-to-noise ratio (SNR) of the averaged interferogram, until the SNR of the averaged interferogram exceeds a SNR threshold. In certain embodiments, determining the SNR includes determining a signal amplitude of a center burst of the averaged interferogram and determining a noise level of the averaged interferogram from data points of the averaged interferogram located away from the center burst. In certain embodiments, determining the SNR includes Fourier transforming the averaged interferogram into a frequency spectrum and numerically integrating the frequency spectrum.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/023,080, filed on Sep. 16, 2020, which claims priority to U.S.Provisional Patent Application No. 62/900,829, filed on Sep. 16, 2019.Each of these applications is incorporated herein by reference in itsentirety.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under grant numbersDE-AR0000539 and DE-FE0029168 awarded by the U.S. Department of Energy.The government has certain rights in the invention.

BACKGROUND

The burning of natural gas emits fewer carbon emissions than the burningof coal, and thus a transition from coal to natural gas may help reduceor revert climate change. The United States is already the world'slargest producer of natural gas, outputting over 37 trillion cubic feetin 2018. In the United States, natural gas represents approximatelyone-third of the nation's entire energy production, the most of anyenergy type. It is also one of the nation's largest energy sources forelectrical generation.

Natural gas is predominantly methane, a potent greenhouse gas. Thepotency of a greenhouse gas is commonly measured by global warmingpotential (GWP), which quantifies how much heat the gas traps in theatmosphere, relative to carbon dioxide, over a specific time horizon. Bydefinition, the GWP of carbon dioxide is one. The GWP of methane is 86over 20 years, and 34 over 100 years.

Significant infrastructure has been constructed, both in the UnitedStates and abroad, to extract, process, transport, and utilize naturalgas. This infrastructure includes wells and rigs for extraction,pipelines and liquid natural gas (LNG) tankers for transportation,liquification and condensation facilities, processing plants forremoving impurities and non-methane components, storage tanks, andindustrial boilers (e.g., refineries, power stations, chemical plants)that utilize methane as an energy source for generating heat.

Since methane is a gas, it can easily escape into the atmosphere throughleaks that form in equipment and components, such as valves, pipes,connectors, pumps, pressure-relief devices, open-ended lines, andsampling connections. Emissions at a typical facility (e.g., refinery orchemical plant) may arise, for example, from seals and gaskets that areimproperly seated or maintained. A typical facility has almost 20,000valves and connectors, and may have over 100,000. Failure of any one ofthese components may result in a leak. However, leaks may also arisefrom corrosion of metal components, as well as damage to components dueto normal wear and tear and/or anomalous operation.

Therefore, to obtain the full environmental benefit of switching fromcoal to natural gas, it is important to reduce the number of methaneleaks and the quantity of methane emitted by each leak. The amount ofleaked methane (also known as “fugitive emissions”) in the United Statesis estimated to be between 1.4% and 2.3% of total production per year.Equivalent to 0.5-0.8 trillion cubic feet, these fugitive emissions areenough to heat between 7 and 11 million homes.

In 2016, the United States Environmental Protection Agency (EPA) passedthree new rules to help reduce methane emissions in the oil and naturalgas industries. These rules include New Source Performance Standardsthat sets emission limits for methane and requires owners/operators ofequipment to find and repair fugitive methane leaks. The EPA estimatesthat these rules will reduce fugitive methane emissions by 510,000 shorttons, or 23 billion cubic feet.

To adhere to the 2016 EPA rules, owners/operators of natural gas wellsites, oil well sites, gathering and boosting stations, and compressorstations must survey their equipment for leaks at fixed schedules.Owners/operators must use optical gas imaging (OGI) to conduct theseleak surveys. The most common type of OGI uses an infrared camera thatis sensitive between 3.3 and 3.4 μm, where methane has absorption lines.However, the performance of an infrared camera depends on weatherconditions (e.g., temperature, wind) as well as the emissivities ofmaterials in the background of the image. As an alternative to OGI,owners/operators may invoke “Method 21” in which surveying is conductedwith a portable instrument, such as an organic vapor analyzer. The 2016EPA rules also allow the EPA to approve the use of emergingleak-monitoring technologies as alternatives to OGI; owners/operatorsmust submit information demonstrating that the alternative technology iscapable of achieving methane reductions equivalent to those that can beachieved when OGI or Method 21 is used to find and repair leaks.

In addition to the oil and gas industries, methane emissions are also ofconcern in agriculture, where global emissions from livestock isestimated at 119 Tg per year (equivalent to 5.9 trillion cubic feet).Other major anthropogenic sources of methane include methane-emittingbacteria that grow in rice paddies (estimated at 115-243 Tg emittedglobally per year), biomass burning (estimated at 40-55 Tg emittedglobally per year), and landfills (estimated at 40-55 Tg emittedglobally per year).

SUMMARY

Many instruments for detecting gas leaks must be located relativelyclose to the source of the leak. For example, an infrared camera usedfor optical gas imaging (OGI) is most efficient at detecting a leak whenlocated within thirty feet of it (with the exception of “super-emitters”that can be easily imaged at somewhat larger distances). For a portableinstrument used under “Method 21” of the 2016 EPA rules, such as anorganic vapor analyzer, the instrument must be even closer, i.e., withina few feet. In either case, proper use of the instrument poses a host ofsafety risks to the operator, especially at an oil and gas facility. Forexample, many potential sources of leaks are in hard-to-access placesthat require the operator to work in confined spaces and/or close tohigh-power machinery and heavy equipment. The operator may be exposed toexcessive heat and noise, as are routinely encountered in oil refineriesand other large industrial complexes. The operator may also need to worknear components operating under high pressure, and thus can explode,resulting in blunt and/or penetrating trauma to the operator. Theoperator may also be exposed to chemicals that can pose a health riskthrough skin contact and/or inhalation (e.g., ethylbenzene, xylene,benzene, toluene, carbon monoxide, sulfur dioxide, hydrogen sulfide,etc.). To reduce this exposure, the operator may need to use arespirator mask and/or other types of personal protective equipment thatlimit their mobility. Furthermore, many leaking gases are flammable andhave low flash points, and thus can cause severe damage if accidentallyignited.

Presented herein is a dual-comb spectroscopy (DCS) system that remotelysenses a variety of trace gases with a sensitivity that rivals, orexceeds, that of prior-art instruments. Advantageously, the DCS systemcan autonomously (i.e., without a human operator) detect a leak morethan one kilometer away from its source, and thus minimize the exposureof human operators to the multitude of safety risks described above.Specifically, when the DCS system determines that no leak exists at aparticular location, a human operator need not be deployed to thelocation to conduct an emission survey. When the DCS system does detecta leak (or another type of abnormal emission), a human operator can, forexample, be deployed to the location with more specific knowledge aboutthe size, location, and composition of the leak, and thus can find andrepair the leak more quickly, minimizing their time at the location.

The DCS system uses two optical frequency combs containing hundreds ofthousands of frequencies, or more, covering the visible, near-infrared,and/or mid-infrared regions of the electromagnetic spectrum. Thus, theDCS system can detect several species of gases, unlike other types ofremote trace gas detectors utilizing single-frequency lasers. Forexample, the DCS system can be configured to detect the most commongases that leak at oil and gas facilities, including methane, acetylene,carbon dioxide, water vapor, carbon monoxide, hydrogen sulfide,ethylene, ethane, propane, butane, and BTEX (benzene, toluene,ethylbenzene, and xylene). In fact, the DCS system can be configured todetect several of these species simultaneously, as well as other typesof volatile organic compounds and hydrocarbons. Furthermore, the DCSsystem detects multiple spectral lines for each species, from which itcan determine a temperature of the leaked gas; knowledge of thetemperature can be helpful for determining the source of the leak.

Embodiments presented herein advantageously improve robustness,reliability, and portability of DCS systems by reducing component count,power consumption, size, and weight. Thus, the embodiments enablefield-deployment of DCS systems while ensuring reliable, continuousoperation throughout a variety of weather conditions that can affectperformance.

In certain embodiments, a data-processing method for an interferogramhaving a plurality of data points advantageously speeds up dataprocessing by removing data points that contain only, or predominantly,noise. The reduction of data points facilitates real-time dataprocessing and reduces the need for high-power offline computingresources. The data-processing method includes recording theinterferogram with a dual frequency comb spectrometer, and gating theinterferogram by retaining a plurality of gated data points of the datapoints, and by discarding a remaining plurality of ungated data points.

In certain embodiments, a method for adaptive dual frequency-combspectroscopy includes repeatedly (i) recording a single interferogramwith a dual frequency-comb spectrometer, (ii) averaging the singleinterferogram into an averaged interferogram, and (iii) determining aSNR of the averaged interferogram, until the SNR of the averagedinterferogram exceeds a SNR threshold.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a functional diagram showing one example of a dual-combspectroscopy (DCS) system that spectroscopically senses trace levels ofa gas, in embodiments.

FIG. 2 shows how a double pulse train is formed by combining first andsecond optical pulse trains generated by respective first and secondoptical frequency combs.

FIG. 3 shows one example of an interferogram obtained from aphotodetector.

FIG. 4 compares frequency components of a local oscillator (LO)frequency comb and a signal (SIG) frequency comb, corresponding to a LOpulse train and a SIG pulse train, respectively, in embodiments.

FIG. 5 is a functional diagram of a dual frequency-comb source (DFCS)that advantageously frequency-stabilizes a first reference laser to atransition in an atomic or molecular vapor, thereby establishing thelaser frequency of the first reference laser as the frequency of thetransition, in embodiments.

FIG. 6 is a functional diagram of a DFCS that is similar to the DFCS ofFIG. 5 , except that the DFCS of FIG. 6 includes one f-2f interferometerfor detecting and stabilizing a carrier-envelope offset (CEO) frequency,in embodiments.

FIG. 7A is a flow chart of a frequency-stabilization method forfield-deployable dual frequency-comb spectroscopy, in embodiments.

FIG. 7B is a flow chart of another frequency-stabilization method forfield-deployable dual frequency-comb spectroscopy, in embodiments.

FIG. 8 shows a temporal sequence of interferograms in which a “walking”center burst is used to determine the laser frequency of the firstreference laser of FIGS. 4 and 5 , in an embodiment.

FIG. 9 is a frequency-domain plot showing the laser frequency f_(A) ofthe first relative to LO and SIG comb teeth of the LO and SIG frequencycombs of FIGS. 4 , in an embodiment.

FIG. 10 is a flow chart of a method for measuring the frequency of alaser with a dual frequency-comb spectrometer, in embodiments.

FIG. 11 is a flow chart of a drift-immune frequency-stabilization methodfor locking a dual frequency-comb spectrometer having first and secondfrequency combs, in embodiments.

FIG. 12 illustrates a method for analyzing an interferogram to quantifythe signal-to-noise ratio (SNR) of the interferogram, in an embodiment.

FIG. 13 illustrates a method for using an upper threshold and a lowerthreshold to identify and reject an interferogram with a signalamplitude too small to provide reliable data, in embodiments.

FIG. 14A illustrates a method for averaging a plurality ofinterferograms together to generate a SNR-enhanced interferogram havinga higher SNR than any one of the interferograms, in embodiments.

FIG. 14B is a flow chart of a method for adaptive dual frequency-combspectroscopy, in embodiments.

FIG. 15 illustrates how a frequency spectrum of an interferogram may bealternatively used to determine the SNR of the interferogram, in anembodiment.

FIG. 16 illustrates a method for gating an interferogram to reduce anumber of data points used for data processing and storage, inembodiments.

FIG. 17 is a flow chart of a data-processing method for an interferogramhaving a plurality of data points, in embodiments.

FIG. 18 shows a sine wave being added to a temporal sequence toadvantageously reduce integral nonlinearity of an analog-to-digitalconverter (ADC) that digitizes the temporal sequence, in embodiments.

FIGS. 19 and 20 show shifted sequences that are similar to a shiftedsequence of FIG. 18 , except that the sine wave of FIG. 18 has a higherfrequency, in embodiments.

FIG. 21 is a functional diagram of an ADC nonlinearity canceler thatadds the sine wave of FIG. 18 to the temporal sequence of FIG. 18 togenerate the shifted sequence of FIG. 18 , digitizes the shiftedsequence into an interferogram sequence, and averages interferograms ofthe interferogram sequence to cancel signal shifts from the sine wave,in embodiments.

FIG. 22 is a flow chart of a method 2200 for improving nonlinearity ofan ADC in a dual frequency-comb spectrometer, in embodiments.

DETAILED DESCRIPTION

FIG. 1 is a functional diagram showing one example of a dual-combspectroscopy (DCS) system 100 that spectroscopically senses trace levelsof a gas 116. Gas 116 may be, for example, methane, acetylene, carbondioxide, or another molecular species. DCS system 100 includes a DCSspectrometer 102 that generates a double pulse train 106 by combiningfirst and second optical pulse trains outputted by respective first andsecond optical frequency combs 120(1), 120(2) (see also first and secondoptical pulse trains 210(1) and 210(2) in FIG. 2 ). Double pulse train106 is delivered, via an optical fiber 112, to an optical transceiver104 mounted on a gimbal mount 110. Optical transceiver 104 couplesdouble pulse train 106 from optical fiber 112 into free space, andgimbal mount 110 is controlled to steer double pulse train 106 toward aretroreflector 118(1) that retroreflects double pulse train 106 into aretroreflected pulse train 108 that propagates back to opticaltransceiver 104. Optical transceiver 104 includes a photodetector thatdetects retroreflected pulse train 108 (see photodetector 220 in FIG. 2) and outputs a corresponding electrical signal to DCS spectrometer 102via a data bus 114. DCS spectrometer 102 includes a data processingmodule 122 that receives and processes the photodetector signal todetermine therefrom the presence of gas 116 along the path traversed bydouble pulse trains 106 and 108.

As described in more detail below, spectra of optical frequency combs120(1) and 120(2) are configured such that spectral components ofoptical pulse train 106 match absorption features of gas 116. Inparticular, several gases of interest for trace gas measurement haveabsorption features between 1.5 and 2.2 μm. For example, the spectra maybe configured such that double pulse train 106 has spectral componentsbetween 176 and 184 THz (i.e., wavelengths between 1.63 and 1.70 μm) formeasuring methane. The spectra may be alternatively configured such thatdouble pulse train 106 has spectral components between 193 and 198 THz(wavelengths between 1.52 and 1.55 μm) for measuring acetylene. Othergases with absorption features between 1.5 and 2.2 μm include watervapor and carbon dioxide.

While these examples describe absorption features in the infrared regionof the electromagnetic spectrum, optical frequency combs 120(1) and120(2) may be configured to measure gases with absorption features inother parts of the electromagnetic spectrum, such as the ultraviolet,visible, near-infrared, mid-infrared, and far-infrared regions. Thus,the term “optical”, as used herein, is not limited to the visible partof the electromagnetic spectrum, and may refer to another region of theelectromagnetic spectrum.

To detect or measure gas leaks that may occur, for example, at an oil orgas facility (e.g., well site, refinery, or chemical plant), a distancebetween optical transceiver 104 and retroreflector 118 may be onekilometer or more. To ensure that retroreflected pulse train 108 returnsto optical transceiver 104, retroreflector 118 may be a corner-cuberetroreflector, a cat's eye retroreflector, a hollow roof prism, oranother type of optic that retroreflects an incident light beamidentically regardless of the direction of the incident light beam. Inembodiments where retroreflector 118 is a planar mirror, retroreflector118 may be actively steered (e.g., via a motorized or piezo-actuatedoptical mount) to correctly retroreflect double pulse train 106 intoretroreflected pulse train 108.

To identify one or more sources of gas 116 (i.e., gas leaks), it isfrequently beneficial to acquire spectroscopic data along several pathsthroughout the spatial region of interest. Thus, several retroreflectors118 may be positioned around the spatial region of interest, as shown inFIG. 1 , and gimbal mount 110 may be controlled to steer double pulsetrain 106 to each of retroreflectors 118. Additional geometries forusing DCS spectrometer 102 with several retroreflectors 118 to obtainspectroscopic data along several paths throughout the spatial region ofinterest may be found in (i) United States Patent ApplicationPublication Number 2016/0334538, titled “Hub and Spoke System forDetecting and Locating Gas Leaks” and published on Nov. 17, 2016, (ii)United States Patent Application Publication Number 2018/0045596, titled“Determining a Location and Size of a Gas Source with a Spectrometer GasMonitor” and published on Feb. 15, 2018, and (iii) InternationalPublication Number WO 2018/075668, titled “Apparatus and Methods forLocation and Sizing of Trace Gas Sources” and published on Apr. 26,2018, all of which are incorporated herein by reference.

In FIG. 1 , DCS spectrometer 102 is advantageously located separate fromoptical transceiver 104 such that DCS spectrometer 102 is not affixed togimbal mount 110. Since DCS spectrometer 102 contains lasers and opticallocks that are sensitive to vibrations and temperature fluctuations, DCSspectrometer 102 may be located in a vessel or room that environmentallyisolates DCS spectrometer 102 from unnecessary mechanical motion (e.g.,movement of gimbal mount 110), wind, precipitation (e.g., wind, ice,snow), and debris (e.g., branches, leaves, insects, bird droppings),thereby helping to improve robustness and reliability of DCSspectrometer 102. For example, DCS spectrometer 102 may be located in avan, building, or mobile office trailer near gimbal mount 110.Alternatively, optical fiber 112 and data bus 114 may be severalkilometers long, wherein DCS spectrometer 102 is located severalkilometers away from gimbal mount 110.

Principles of Dual-Comb Spectroscopy

FIG. 2 shows how double pulse train 106 is formed by combining first andsecond optical pulse trains 210(1), 210(2) generated by respective firstand second optical frequency combs 120(1), 120(2) of FIG. 1 . Forclarity in the following description, first optical frequency comb120(1) and corresponding first optical pulse train 210(1) may bereferred to as “local oscillator” (LO) frequency comb 120(1) and LOpulse train 210(1), respectively, while second optical frequency comb120(2) and second optical pulse train 210(2) may be referred to as“signal” (SIG) frequency comb 120(2) and SIG pulse train 210(2),respectively. SIG pulse train 210(2) is formed from a sequence of SIGpulses 204 that repeat with a repetition period T_(rep) ^((SIG))), whileLO pulse train 210(1) is formed from a sequence of LO pulses 206 thatrepeat with a repetition period) T_(rep) ^((LO)) that is greater thanrepetition period T_(rep) ^((SIG)) by a difference ΔT that is smallcompared to both) T_(rep) ^((LO)) and T_(rep) ^((SIG)), and that issmaller than, or comparable to, a temporal pulse width of each of pulses204 and 206. In FIG. 2 , LO pulse train 210(1) is shown as a solid line,and SIG pulse train 210(2) is shown as a dashed line.

In FIG. 2 , a first SIG pulse 204(1) and a first LO pulse 206(1) aredetected simultaneously by a photodetector 220. Due to the difference ΔTof repetition periods, a second LO pulse 206(2) lags a second SIG pulse204(2) by ΔT, a third LO pulse 206(3) lags a third SIG pulse 204(3) by2ΔT, and so on. The difference ΔT may be chosen such that T_(rep)^((SIG)) is an integer multiple of ΔT, i.e., T_(rep) ^((SIG))=m ΔT foran integer m. Thus, the lag between the m^(th) LO pulse 206 and thecorresponding SIG pulse 204 is T_(rep) ^((SIG)), in which case them^(th) LO pulse 206 is detected simultaneously with the (m+1)th SIGpulse 204, similar to first LO pulse 206(1) and first SIG pulse 204(1).Equivalently, double pulse train 106 is periodic with a periodT_(dpt)=(m−1)T_(rep) ^((LO))=mT_(rep) ^((SIG)), and during one periodT_(dpt), SIG pulse train 210(2) generates m SIG pulses 204, and LO pulsetrain 210(1) generates m−1 LO pulses 206.

Photodetector 220 responds to the total intensity of pulse trains220(1), 220(2) impinging thereon, including cross-terms that arise frominterference between LO pulses 206 and SIG pulses 204. Thus,photodetector 220 outputs a photodetector signal that is a measure ofthe cross-correlation between pulse trains 210(1), 210(2). When an LOpulse and a SIG pulse coincide (i.e., have a large temporal overlap) atphotodetector 220 (e.g., SIG pulse 204(1) and LO pulse 206(1)),constructive interference between the LO pulse and the SIG pulsegenerates a relatively large photodetector signal. When the LO pulse andthe SIG pulse do not coincide (i.e., have a small temporal overlap) atphotodetector 220 (e.g., SIG pulse 204(5) and LO pulse 206(5)),destructive interference between the LO pulse and the SIG pulse resultsin a small photodetector signal.

FIG. 3 shows one example of an interferogram 302 obtained fromphotodetector 220. Interferogram 302 contains m−1 data points that areobtained by sampling the output of photodetector 220 synchronously withm−1 consecutive LO pulses 206. Thus, in the laboratory, interferogram302 is acquired in a frame time T_(f)=(m−1)T_(rep) ^((LO)), which isshown in FIG. 3 relative to an initial time chosen to be zero. This“laboratory” time is denoted by t. Interferogram 302 contains a signalcomponent 306, or “center burst”, and a noise component 308.

FIG. 3 also shows a zoomed-in section 304 of center burst 306. Insection 304, data points 310, equally spaced in time by a time step 314,are connected by a dashed line 312 to guide the eye. In laboratory time,time step 314 equals T_(rep) ^((LO)). However, time step 314 also equalstime lag ΔT that accrues between LO pulses 206 and SIG pulses 204 (seeFIG. 2 ). Thus, the time axis of interferogram 402 may be scaled byΔT/T_(rep) ^((LO)) to switch from laboratory time to an “equivalent”time denoted by τ. In equivalent time, interferogram 302 represents theelectric field of signal pulse train 210(1) as sampled by LO pulse train210(1) with a time spacing of ΔT. Thus, in equivalent time,interferogram 302 corresponds to a duration of τ_(f)=(m−1)×ΔT.

The output of photodetector 220 repeats with a period equal to frametime T_(f) (in laboratory time), and therefore interferogram 302 alsorepeats with the frame time T_(f). As a result, several interferogramsmay be sequentially measured and averaged together to improvesignal-to-noise ratio (SNR), provided that the total time to measure theseveral interferograms is less than the mutual coherence time offrequency combs 120(1), 120(2). Furthermore, the Fourier transform ofinterferogram 302 yields an absorption spectrum and/or optical transferfunction of gas 116 from which a path-integrated quantity of gas 116 maybe estimated.

FIG. 4 compares frequency components of LO frequency comb 120(1) and SIGfrequency comb 120(2), corresponding to LO pulse train 210(1) and SIGpulse train 210(2), respectively. LO frequency comb 120(1) includes aplurality of coherent LO frequency components 406, or LO comb teeth 406,that are equally-spaced in frequency by a LO repetition rate f_(rep)^((LO))=1/T_(rep) ^((LO)). The LO repetition rate f_(rep) ^((LO)) mayalso be referred to herein as a LO comb spacing. LO comb teeth 406 arealso offset from zero frequency by a LO carrier-envelope offset (CEO)frequency f₀ ^((LO)) that has a value such that −f_(rep) ^((LO))/2<f₀^((LO))≤f_(rep) ^((LO))/2. Frequencies of LO comb teeth 406 may berepresented mathematically by f_(n) ^((LO))=nf_(rep) ^((LO))+f₀ ^((LO)),where positive integer n indexes the frequencies f_(n) ^((LO)).Similarly, SIG frequency comb 120(2) includes a plurality of coherentSIG frequency components 404, or SIG comb teeth 404, that areequally-spaced in frequency by a SIG repetition rate f_(rep)^((SIG))=1/T_(rep) ^((SIG))=1/(T_(rep) ^((LO))−ΔT). The SIG repetitionrate f_(rep) ^((LO)) may also be referred to herein as a SIG combspacing. SIG comb teeth 404 are also offset from zero frequency by a SIGCEO frequency f₀ ^((SIG)) with a value such that −f_(rep) ^((SIG))/2<f₀^((SIG))≤f_(rep) ^((SIG))/2. Frequencies of SIG comb teeth 404 may berepresented mathematically by f_(n′) ^((SIG))=n′f_(rep) ^((SIG))+f₀^((SIG)), where positive integer n′ indexes the frequencies f_(n′)^((SIG)).

To measure or detect gas 116 with DCS system 100, a value of ΔT isselected and implemented by controlling repetition rates f_(rep) ^((LO))and f_(rep) ^((SIG)) and CEO frequencies f₀ ^((LO)) and f₀ ^((SIG)). Asdescribed above, the value of ΔT may be chosen such that ΔT dividesT_(rep) ^((SIG)) by an integer m, wherein m−1 equals the number of datapoints in one interferogram. To ensure that the interferograms repeatevery m LO pulses 206, comb teeth 404 and/or 406 are controlled toestablish lower and upper anchor frequencies. At the lower anchorfrequency, the frequency of one LO tooth 406 equals the frequency of oneSIG tooth 404. At the upper anchor frequency, the frequency of anotherLO tooth 406 equals the frequency of another SIG tooth 404. Frequencycombs 120(1), 120(2) should have no matching teeth 404, 406 atfrequencies between the lower and upper anchor frequencies, which may beachieved by selecting f_(rep) ^((LO)) and/or f_(rep) ^((SIG)) such thatthe number of LO teeth 406 between the lower and upper anchorfrequencies is one more than the number of SIG teeth 404 between thelower and upper anchor frequencies. The frequency range between upperand lower anchor frequencies is also be referred to herein as a “Nyquistwindow”. The anchor frequencies should be chosen so that all spectralfeatures of gas 116 to be detected fall within the Nyquist window.

Comb teeth 404, 406 may be controlled (e.g., via phase or frequencylocking) to establish mutual coherence between pulse trains 210(1) and210(2), thereby ensuring that photodetector 220 measures thecross-correlation between pulse trains 210(1), 210(2) rather thanincoherent noise/jitter. Alternatively, jitter between pulse trains210(1), 210(2) may be measured and used to correct interferogramsobtained when frequency combs 120(1), 120(2) are not activelystabilized.

It is beneficial to frequency-stabilize LO teeth 406 and SIG teeth 404to ensure that the optical phase of SIG pulse train 210(2) sampled by LOpulses 206 does not drift between a data point of one interferogram andthe same data point of the next interferogram. It is additionallybeneficial to absolutely frequency-stabilize comb teeth 404 and 406 sothat spectral features obtained from the Fourier transform of aninterferogram may be more easily compared to known absorption lines ofgas 116, thereby facilitating the identification of gas 116.

To establish mutual coherence between frequency combs 120(1) and 120(2),and to stabilize teeth 404 and 406, frequency combs 120(1) and 120(2)may be stabilized to first and second reference lasers 410(1) and410(2). Stabilizing LO frequency comb 120(1) means stabilizing thefrequencies of all LO teeth 406 by controlling the LO CEO frequency f₀^((LO)) and the LO repetition rate f_(rep) ^((LO)) (or LO comb spacing).Similarly, stabilizing SIG frequency comb 120(2) means stabilizing thefrequencies of all SIG teeth 404 by controlling the SIG CEO frequency f₀^((SIG)) and the SIG repetition rate f_(rep) ^((SIG)) (or SIG combspacing).

As shown in FIG. 4 , first reference laser 410(1) outputs first coherentlight (see first coherent light 560 in FIG. 5 ) at a reference frequencyf_(A) that is closest in frequency to a LO tooth 406(1). Heterodyningthe first coherent light and LO frequency comb 120(1) generates a firstbeat whose frequency may be locked, via a phase-lock loop, to a firstfrequency offset Δf_(A) ^((LO)) by controlling the LO CEO frequency f₀^((LO)). This stabilizes the frequency of LO tooth 406(1) with afrequency stability and phase noise determined by first reference laser410(1).

Second reference laser 410(2) outputs second coherent light (see secondcoherent light 570 in FIG. 5 ) at a reference frequency f_(B), differentfrom f_(A), that is closest in frequency to a LO tooth 404(2).Heterodyning the second coherent light and LO frequency comb 120(1)generates a second beat whose frequency may be locked, via a phase-lockloop, to a second frequency offset Δf_(B) ^((LO)) by controlling the LOrepetition rate f_(rep) ^((LO)). This stabilizes the frequency of LOtooth 406(2) with a frequency stability and phase noise determined bysecond reference laser 410(2).

Reference frequency f_(A) is also closest in frequency to a SIG tooth404(1). Heterodyning the first coherent light and SIG frequency comb120(2) generates a third beat whose frequency may be locked, via aphase-lock loop, to a third frequency offset Δf_(A) ^((SIG)) bycontrolling the SIG CEO frequency f₀ ^((SIG)) This stabilizes thefrequency of SIG tooth 404(1) with a frequency stability and phase noisedetermined by first reference laser 410(1).

Similarly, reference frequency f_(B) is also closest in frequency to aSIG tooth 404(2). Heterodyning the second coherent light and SIGfrequency comb 120(2) generates a fourth beat whose frequency may belocked, via a phase-lock loop, to a fourth frequency offset Δf_(B)^((SIG)) by controlling the SIG repetition rate f_(rep) ^((SIG)). Thisstabilizes the frequency of SIG tooth 404(2) with a frequency stabilityand phase noise determined by second reference laser 410(2).

When frequency combs 120(1) and 120(2) are tightly locked to first andsecond reference lasers 410(1), 410(2), mutual coherence is establishedbetween frequency combs 120(1) and 120(2). Here, “tightly locked” meansthat each of the phase-lock loops has a loop bandwidth and gain highenough to replace the intrinsic phase/frequency noise of frequency combs120(1) and 120(2) with the phase/frequency noise (i.e., laser linewidth)of reference lasers 410(1) and 410(2) such that the phase/frequencynoise of frequency combs 120(1), 120(2) is dominated by that ofreference lasers 410(1) and 410(2).

When frequency combs 120(1) and 120(2) are locked to first and secondreference lasers 410(1), 410(2), the frequency stability of any one ofteeth 404 and 406 is determined by a combination of the frequencystabilities of reference lasers 410(1) and 410(2). Thus, teeth 404 and406 will drift in frequency as reference frequencies f_(A) and f_(B)drift. Many laser systems that can serve as first and second referencelasers 410(1) and 410(2) do not have sufficient levels of intrinsicfrequency stability for DCS, and may be actively frequency-stabilized toreduce this drift to acceptable levels for DCS.

Simplified DCS Spectrometer for Portable Remote Gas Sensing

To properly identify a trace gas, frequencies of teeth 404, 406 must beknown. That is, optical frequency combs 120(1) and 120(2) must becalibrated for accuracy. A Fourier transform of interferogram 302generates a series of frequency-data points that are spaced in frequencyby Δf_(rep)=f_(rep) ^((SIG))−f_(rep) ^((LO)), which can be accuratelymeasured with a frequency counter. The frequency-data points are offsetin frequency by the lower anchor frequency, which can be determined fromreference frequencies f_(A) and/or f_(B), and frequency offsets Δf_(A)^((SIG)), Δf_(A) ^((LO)), Δf_(B) ^((SIG)), and/or Δf_(B) ^((LO)). Sincethe frequency offsets are known, calibration of optical frequency combs120(1), 120(2) is most challenged by accurately determining laserfrequencies f_(A) and f_(B). Laser frequencies f_(A) and f_(B) can bedetermined with one of optical frequency combs 120(1) and 120(2), butonly if f_(A) and f_(B) are known to within the comb spacing f_(rep). Tomeasure f_(A) and f_(B) to this level of precision requires ahigh-performance wavemeter that increases the size, weight, andcomplexity of DCS system 100, and thus inhibits portability of DCSsystem 100.

FIG. 5 is a functional diagram of a dual frequency-comb source (DFCS)500 that advantageously frequency-stabilizes first reference laser410(1) to a transition in an atomic or molecular vapor, therebyestablishing the laser frequency f_(A) as the frequency of thetransition. The transition frequency, based on the atomic or molecularspecies of the vapor, is known with sufficient accuracy that opticalfrequency combs 120(1) and 120(2) can be calibrated without ahigh-performance wavemeter. Given that (i) the spectral features to bemeasured in gas 116 are Doppler-broadened to widths of severalgigahertz, and (ii) the comb spacings f_(rep) of optical frequency combs120(1) and 120(2) are typically greater than 50 MHz, the laser frequencyf_(A) only needs to be stabilized and accurately determined to within 25MHz. An accuracy of just a few megahertz can be readily achieved bylocking f_(A) to any of several atomic and molecular species in arobust, compact vapor-cell-based spectrometer, even over a wide varietyof environmental and operating parameters (e.g., temperature of thevapor cell, laser power probing the vapor, etc.). DFCS 500 may be usedwith DCS spectrometer 102 of FIG. 1 .

Advantageously, DFCS 500 stabilizes all comb teeth 404, 406 of bothfrequency combs 120(1), 120(2) without f-2f interferometers typicallyused to detect CEO frequencies f₀ ^((LO)) and f₀ ^((SIG)). Thus, DFCS500 does not require each of frequency combs 120(1), 120(2) to span morethan an octave in frequency, thereby relaxing the requirements for theirgeneration. By operating without f-2f interferometers, DFCS 500eliminates two optical amplifiers, two nonlinear optical fibers, and twofrequency-doubling crystals, among other components, advantageouslyimproving portability and reliability by reducing component count andpower consumption.

DFCS 500 includes a LO comb laser 510 that generates a first opticalpulse train 540 that is amplified by a first optical amplifier 514(1)into a first amplified pulse train 542. DFCS 500 also includes a SIGcomb laser 530 that generates a second optical pulse train 550 that isamplified by a second optical amplifier 514(2) into a second amplifiedpulse train 552. LO comb laser 510 and SIG comb laser 530 may each beany type of pulsed laser whose output train has a frequency-comb-likestructure. For example, LO comb laser 510 and SIG comb laser 530 mayeach be a solid-state pulsed laser, such as a Kerr-lens, mode-locked,titanium:sapphire laser. Alternatively, LO comb laser 510 and SIG comblaser 530 may each be a mode-locked fiber laser based on a fiber dopedwith erbium, ytterbium, holmium, or thulium. Alternatively, LO comblaser 510 and SIG comb laser 530 may each be a mode-locked diode laser,mode-locked disc laser, or another type of mode-locked laser. When LOcomb laser 510 and SIG comb laser 530 are mode-locked pulsed lasers, LOcomb laser 510 and SIG comb laser 530 may be implemented with activemode-locking, passive mode-locking, or a hybrid thereof.

DFCS 500 also includes first reference laser 410(1) that outputs a firstcoherent light 560 at reference frequency f_(A), and second referencelaser 410(2) that output second coherent light 570 at referencefrequency f_(B). First and second reference lasers 410(1), 410(2) may beany type of tunable, single-frequency laser whose output has a frequencythat overlaps with first and second frequency combs 120(1), 120(2) forheterodyning therewith. For example, first and second reference lasers410(1), 410(2) may each be a continuous-wave (cw) diode laser (e.g.,external-cavity diode laser, distributed Bragg reflector (DBR) diodelaser, distributed feedback laser (DFB), vertical-cavitysurface-emitting laser (VCSEL), quantum cascade laser, etc.), fiberlaser (e.g., erbium-doped, ytterbium-doped, etc.), solid-state laser(e.g., Nd:YAG laser), or gas laser (e.g., He—Ne laser). First referencelaser 410(1) may include a frequency-doubling crystal (e.g., potassiumtitanyl phosphate (KTP), lithium niobate (LiNbO₃), potassium niobate(KNbO₃), etc.), or another type of nonlinear optical element, so thatfrequency f_(A) of first coherent light 560 overlaps with first andsecond frequency combs 120(1), 120(2) for heterodyning therewith; secondreference laser 410(2) may also include a frequency-doubling crystal forsimilar reasons.

DFCS 500 also includes a first laser stabilizer 536 that samples firstcoherent light 560 and controls first reference laser 410(1) tostabilize the frequency f_(A). First laser stabilizer 536 may include avapor-cell spectrometer used to generate a feedback signal for lockingf_(A) to a transition in the vapor. In one embodiment, the vaporspectrometer includes a vapor cell containing an atomic or molecular gasprobed with first coherent light 560 in a Doppler-free configuration(i.e., overlapping, counterpropagating probe and pump beams) to generatea dispersive signal, at a transition in the atomic or molecular species,that is fed back to first reference laser 410(1) to stabilize f_(A) tothe transition frequency. In another embodiment, first laser stabilizer536 is a phase-lock loop that phase-locks first reference laser 410(1)to another reference laser (not shown in FIG. 5 ) whose output is morefrequency-stable than f_(A) when first reference laser 410(1) isfree-running. In another embodiment, first laser stabilizer 536 is aphase-lock loop that phase-locks first reference laser 410(1) to a thirdfrequency comb (not shown in FIG. 5 ) whose frequency components aremore frequency-stable than f_(A) when first reference laser 410(1) isfree-running. The third frequency comb may be self-referenced and lockedto a RF frequency standard such that all frequency components of thethird frequency comb are accurate relative to the RF frequency standard.First laser stabilizer 536 may also include a frequency-doubling crystal(e.g., KTP, LiNbO₃, KNbO₃, etc.), or another type of nonlinear opticalelement, such that frequency f_(A) is suitable for generating adispersive feedback signal (e.g., to resonate with an atomic transitionin a vapor). First laser stabilizer 536 may be another type ofoptical/electrical device that controls first reference laser 410(1) tostabilize f_(A) without departing from the scope hereof.

DFCS 500 includes: (i) a first photodetector 512(1) that samples firstamplified pulse train 542 for measuring the LO repetition rate f_(rep)^((LO)), (ii) a second photodetector 512(2) that detects the first beatbetween first amplified pulse train 542 and first coherent light 560,(iii) a third photodetector 512(3) that detects the third beat betweensecond amplified pulse train 552 and first coherent light 560, (iv) afourth photodetector 512(4) that detects the second beat between firstamplified pulse train 542 and second coherent light 570, (v) a fifthphotodetector 512(5) that detects the fourth beat between secondamplified pulse train 552 and second coherent light 570, and (vi) asixth photodetector 512(6) that samples second amplified pulse train 552for measuring the SIG repetition rate f_(rep) ^((SIG)).

DFCS 500 also includes a LO comb stabilizer 542 that locks the firstbeat to a selected value of Δf_(A) ^((LO)) controlling LO comb laser 510to change the CEO frequency f₀ ^((LO)). LO comb stabilizer 542 alsolocks the second beat to a selected value of Δf_(B) ^((LO)) bycontrolling LO comb laser 510 to change the repetition rate f_(rep)^((LO)). For example, LO comb stabilizer 542 may phase-lock the firstbeat to Δf_(A) ^((LO)) by controlling a pump that powers LO comb laser510, and phase-lock the second beat to Δf_(B) ^((LO)) by controlling acavity spacing of LO comb laser 510. The lock frequencies Δf_(A) ^((LO))and Δf_(B) ^((LO)) are RF frequencies that may be generated from acommon RF frequency reference (e.g., quartz oscillator, microwave atomicclock, atomic frequency standard, maser, etc.) using frequency synthesistechniques known in the art.

DFCS 500 also includes a SIG comb stabilizer 532 that locks the thirdbeat to a selected value of Δf_(A) ^((SIG)) by controlling SIG comblaser 530 to change the CEO frequency f₀ ^((SIG)). SIG comb stabilizer532 also locks the fourth beat to a selected value of Δf_(B) ^((SIG)) bycontrolling SIG comb laser 530 to change the repetition rate f_(rep)^((SIG)). For example, SIG comb stabilizer 532 may phase-lock the thirdbeat to Δf_(A) ^((SIG)) by controlling a pump that powers SIG comb laser530, and phase-lock the fourth beat to Δf_(B) ^((SIG)) by controlling acavity spacing of SIG comb laser 530. The lock frequencies Δf_(A)^((SIG)) and Δf_(B) ^((SIG)) are RF frequencies that may be synthesizedfrom the same frequency reference used to generate Δf_(A) ^((LO)) andΔf_(B) ^((LO)).

DFCS 500 also includes a second laser stabilizer 534 that uses therepetition rate f_(rep) ^((LO)) as measured by first photodetector510(1), to control second reference laser 410(2) to stabilize frequencyf_(B). When LO comb laser 510 is phase-locked to second reference laser410(2) via control of the LO repetition rate f_(rep) ^((LO)), drift offrequency f_(B) is detectable as a change in f_(rep) ^((LO)). Secondlaser stabilizer 534 compares the measured f_(rep) ^((LO)) to a chosenlocking frequency to generate a frequency correction, and controlssecond reference laser 410(2) according to the frequency correction tochange f_(B) so that the measured f_(rep) ^((LO)) equals the chosenlocking frequency, thereby stabilizing frequency f_(B). The chosen lockfrequency may be frequency synthesized from the same frequency referenceused to generate Δf_(A) ^((LO)), Δf_(B) ^((LO)), Δf_(A) ^((SIG)) andΔf_(B) ^((SIG)).

In certain embodiments, DFCS 500 includes a first nonlinear opticalfiber 520(1) that spectrally broadens first amplified pulse train 542such that LO frequency comb 120(1) has a spectral width that coversspectral features (i.e., absorption lines) of interest in gas 116.Similarly, DFCS 500 may include a second nonlinear optical fiber 520(2)that spectrally broadens second amplified pulse train 532 such that SIGfrequency comb 120(2) also has a spectral width that covers the spectralfeatures of interest. Each of nonlinear optical fibers 520(1), 520(2)may be any type of nonlinear optical element that spectrally broadens anoptical pulse train (e.g., amplified pulse trains 542, 532) whilepreserving the coherent comb-like structure of the optical pulse train.For example, each of nonlinear optical fibers 520(1), 520(2) may be ahighly nonlinear optical fiber, an air-silica microstructure fiber, aphotonic crystal fiber, or another type of nonlinear optical elementthat uses self-phase modulation and/or four-wave mixing to implementspectral broadening.

While FIG. 5 shows first nonlinear optical fiber 520(1) afterphotodetectors 512(1), 512(2), and 512(4), first nonlinear optical fiber520(1) may be alternatively located before any of photodetectors 512(1),512(2), and 512(4). Such an alternative location may be beneficial ifthe spectral broadening introduced by first nonlinear optical fiber520(1) improves the SNR of the detected heterodyne beat(s). Similararguments hold for the location of second nonlinear optical fiber 520(2)relative to photodetectors 512(3), 512(4), and 512(6).

As shown in FIG. 5 , DFCS 500 may include first and second opticalfilters 522(1), 522(2) that filter respective LO and SIG frequency combs120(1), 120(2) to block comb teeth 404, 406 lying outside of thespectral region of interest (i.e., Nyquist window) that may causealiasing of the interferogram. Thus, each of first and second opticalfilters 522(1), 522(2) may block comb teeth 404, 406 with frequenciesbelow the lower anchor frequency and above the upper anchor frequency.

In some embodiments, DFCS 500 is configured without one or both ofoptical amplifiers 514(1), 514(2) when one or both of LO comb laser 510and SIG comb laser 530 outputs a frequency comb of sufficient spectralwidth and power for detecting gas 116. Optical amplifiers 514(1), 514(2)boost the optical power of respective optical pulse trains 540, 530 toimprove spectral broadening in corresponding nonlinear optical fibers520(1), 520(2). Thus, in embodiments where nonlinear optical fibers520(1), 520(2) are not included in DFCS 500, optical amplifiers 514(1),512(2) may not be needed, advantageously improving portability byreducing component count and power consumption, and advantageouslyimproving reliability by removing components with limited operationallifetimes (i.e., optical amplifiers).

The accuracy and stability of any one of comb teeth 404, 406 ultimatelyderives from (i) the accuracy and stability of the RF frequencyreference used to generate Δf_(A) ^((LO)), Δf_(B) ^((LO)), Δf_(A)^((SIG)), Δf_(B) ^((SIG)), and the lock frequency for locking f_(rep)^((SIG)), and (ii) the accuracy and stability of first reference laser410(1), when locked via first laser stabilizer 536. The RF frequencyreference may be linked to the cesium second (e.g., via locking to a GPSsignal or cesium atomic standard), thereby ensuring high accuracy of allRF locking frequencies. The accuracy of first reference laser 410(1) maybe determined via calibration, or previous knowledge of the transitionfrequency of the atomic or molecular transition to which first referencelaser 410(1) is locked.

As an example of first reference laser 410(1) and first laser stabilizer536, first reference laser 410(1) may be a diode laser at 1560 nm, andfirst laser stabilizer 536 may include a frequency-doubling crystal thatconverts the light at 1560 nm into light at 780 nm. First laserstabilizer 536 may also include a rubidium vapor-cell spectrometer forlocking the frequency-doubled light to the D₂ transition in rubidium.The frequency of this transition is known with an accuracy of a few kHz,and a compact rubidium vapor-cell spectrometer can be used to stabilizea laser frequency to within 1 MHz of the transition frequency. Firstlaser stabilizer 536 may lock the frequency f_(A) to the transitionusing any technique known in the art, including saturated absorptionspectroscopy, first-derivative locking, third-derivative locking,frequency-modulation spectroscopy, modulation transfer spectroscopy,polarization spectroscopy, and dichroic atomic vapor laser locking(DAVLL).

As another example, first reference laser 410(1) may be a diode laseroutputting first coherent light 560 at 1704 nm, wherein first laserstabilizer 536 includes a frequency-doubling crystal that converts firstcoherent light 560 into frequency-doubled light at 852 nm. First laserstabilizer 536 may also include a cesium vapor cell spectrometer forlocking the frequency-doubled light to the D₂ transition in cesium.Alternatively, first laser stabilizer 536 may contain a vapor cell ofgas 116 (e.g., methane), wherein first last stabilizer 536 stabilizesthe frequency f_(A) of first reference laser 410(1) to a transition ingas 116 without frequency doubling. Other atomic and molecular speciesthat may be used to form a vapor cell spectrometer include potassium(K), argon (Ar), krypton (Kr), iodine (I₂), tellurium (Te₂), water vapor(H₂O), ethane (C₂H₆), acetylene (C₂H₆), carbon monoxide (CO), carbondioxide (CO₂), and hydrogen cyanide (HCN).

In FIG. 5 , first and second amplified pulse trains 542, 552 are splitinto different portions for detecting various heterodyne beats andgenerating pulse trains 210(1), 210(2). When gas 116 is detected basedon spectral features in the infrared (e.g., 1.3-1.8 microns), DFCS 500may be implemented with all fiber-optic components. In this case, firstand second amplified pulse trains 542, 552 may be split via fiber-opticbeamsplitters. Alternatively, first and second amplified pulse trains542, 552 may be split via free-space beamsplitters (e.g., plates, cubes)when first and second amplified pulse trains 542, 552 propagate throughfree space.

First optical pulse train 210(1) (corresponding to LO frequency comb120(1)) and second optical pulse train 210(2) (corresponding to SIGfrequency comb 120(2)) may be combined (via a beam splitter or beamcombiner) into one optical beam to form double pulse train 106, as shownin FIG. 5 . Double pulse train 106 may then be coupled into opticalfiber 112 for delivery to optical transceiver 104, as shown in FIG. 1 .

In some embodiments, second reference laser 410(2) isfrequency-stabilized similarly to first reference laser 410(1). Forexample, second reference laser 410(2) may also be locked to an atomicor molecular transition, via measurement in a vapor cell spectrometer.In these embodiments, second laser stabilizer 534 samples secondcoherent light 570 similarly to how first laser stabilizer 534 samplesfirst coherent light 560. Second laser stabilizer 534 then controlssecond reference laser 410(2) to stabilize the frequency f_(B) of secondreference laser 410(2) to the atomic or molecular transition probed inthe vapor cell spectrometer.

In embodiments where both first and second reference lasers 410(1),410(2) are stabilized to atomic/molecular transitions, first and secondlaser stabilizers 410(1), 410(2) may be combined to use one vapor cell,advantageously reducing component count and eliminating a possiblesource of differential frequency drift between f_(A) and f_(B). Forexample, when first reference laser 410(1) is a diode laser at 1560 nmwhose output is frequency-doubled for locking to the D₂ transition inrubidium at 780 nm, second reference laser 410(2) may be a diode laserat 1590 nm whose output is frequency-doubled for locking to the D₁transition in rubidium at 795 nm. Both transitions may be probed bypassing both frequency-doubled beams through one rubidium vapor cell.The vapor cell may be filled with two species of vapor such that firstand second reference lasers 410(1), 410(2) are stabilized to transitionsin different species. For example, when first reference laser 410(1) isa diode laser at 1560 nm whose output is frequency-doubled for lockingto the D₂ transition in rubidium at 780 nm, second reference laser410(2) may be a diode laser at 1704 nm whose output is frequency-doubledfor locking to the D₂ transition in cesium at 852 nm. Other combinationsof vapors may be used without departing from the scope hereof.

While FIG. 5 shows second laser stabilizer 534 controlling secondreference laser 410(2) to stabilize the frequency f_(B) via measurementsof the LO repetition rate f_(rep) ^((LO)), second laser stabilizer 534may be alternatively configured to stabilize the frequency f_(B) viameasurements of the SIG repetition rate f_(rep) ^((SIG)). Specifically,when SIG comb laser 530 is phase-locked to second reference laser 410(2)via control of the SIG repetition rate f_(rep) ^((SIG)), drift offrequency f_(B) is detectable as a change in f_(rep) ^((SIG)) measuredby sixth photodetector 512(6). Second laser stabilizer 534 compares themeasured f_(rep) ^((SIG)) to a chosen locking frequency to generate afrequency correction, and controls second reference laser 410(2)according to the frequency correction to change f_(B) so that themeasured f_(rep) ^((SIG)) equals the chosen locking frequency, therebystabilizing frequency f_(B).

FIG. 6 is a functional diagram of a DFCS 600 that is similar to DFCS 500of FIG. 5 , except that DFCS 600 includes one f-2f interferometer 604for detecting and stabilizing the CEO frequency f₀ ^((LO)).Advantageously, DFCS 600 stabilizes all comb teeth 404, 406 to thecommon RF frequency reference using only one f-2f interferometer 604.DFCS 600 operates without first reference laser 410(1) being locked toan atomic or molecular transition.

To detect to f₀ ^((LO)) with f-2f interferometer 604, DFCS 600 includesan optical amplifier 614 and nonlinear optical fiber 620 for broadeningthe spectrum of first amplified pulse train 542 to extend over anoctave. Alternatively, f-2f interferometer 604 may sample firstamplified pulse train 542 after spectral broadening in nonlinear opticalfiber 520(1). Although not shown in FIG. 6 , f-2f interferometer 604 mayinclude a frequency-doubling crystal configured for doubling thefrequencies of LO comb teeth 406 covering a low-frequency portion of LOfrequency comb 120(1). Furthermore, f-2f interferometer 604 may includean additional photodetector (similar to any of photodetectors 512) fordetecting a heterodyne beat between the frequency-doubled low-frequencyportion and a high-frequency portion of LO frequency comb 120(1). Thelowest-frequency beat detected by this additional photodetector is f₀^((LO)).

DFCS 600 includes an LO comb stabilizer 642 that is similar to LO combstabilizer 542 of DFCS 500, except that LO comb stabilizer 642phase-locks f₀ ^((LO)) to a selected lock frequency. The lock frequencymay be generated from the common RF frequency reference. LO combstabilizer 642 phase locks the first beat, detected by photodetector512(2), to a selected value of Δf_(A) ^((LO)) by controlling LO comblaser 510 to change the repetition rate f_(rep) ^((LO)). DFCS 600 alsoincludes a first laser stabilizer 636 that measures the repetition ratef_(rep) ^((LO)) detected by photodetector 512(1), and controls firstreference laser 410(1) to change f_(A) so that f_(rep) ^((LO)) is lockedto a desired repetition rate. That is, first laser stabilizer 636stabilizes f_(A) similarly to how second laser stabilizer 534 stabilizesfrequency f_(B) of second reference laser 410(2) in FIG. 5 .

With LO comb stabilizer 642 and first laser stabilizer 636, all LO combteeth 406 are locked with respect to the common RF frequency reference.When the common frequency reference is traceable to the cesium second,the frequencies of all LO teeth 406 have absolute accuracy. Furthermore,locking f_(rep) ^((LO)) by controlling f_(A) advantageously transfersthe narrow linewidth of first coherent light 560 to LO teeth 406, whichmay be 10 kHz, or less. By comparison, when f_(rep) ^((LO)) isphase-locked by directly controlling LO comb laser 510, the phase noiseof one LO tooth 406 will equal the phase noise of the common RFfrequency reference, scaled up by the ratio of the frequency of the oneLO tooth 406 to the frequency of the reference (e.g., 10 MHz). Thisscale factor, which may be 107 or more, is so large that the one LOtooth 406 will be significantly phase/frequency-broadened, reducingspectral sensitivity.

To frequency-stabilize SIG comb teeth 404, DFCS 600 includes a secondlaser stabilizer 634 that phase-locks second reference laser 410(2) toLO tooth 406(2) using the second beat detected by photodetector 512(4).Thus, second laser stabilizer 634 transfers the phase noise andfrequency stability of LO tooth 406(2) to frequency f_(B). DFCS 600 alsoincludes SIG comb stabilizer 532, which controls signal comb laser 530to phase-lock the third beat, detected by photodetector 512(3), toΔf_(A) ^((SIG)), and which controls signal comb laser 530 to phase-lockthe fourth beat, detected by photodetector 512(5), to Δf_(B) ^((SIG)).Thus, SIG comb stabilizer 532, by locking SIG comb laser 530 to firstand second reference lasers 410(1), 410(2), transfers the accuracy andfrequency stability of the common RF frequency reference to all SIGteeth 404.

FIG. 7A is a flow chart of a frequency-stabilization method 700 forfield-deployable dual frequency-comb spectroscopy. Method 700frequency-stabilizes teeth of first and second frequency combs relativeto a single RF frequency reference and a frequency-stabilized firstreference laser. Method 700 also establishes coherence between the firstand second frequency combs based on coherence of the first and secondreference lasers. Method 700 may be implemented with DFCS 500 of FIG. 5.

Method 700 includes steps 708, 710, 712, and 714, which may occur in anyorder. In step 708, a first tooth, of a first frequency comb of teethequally-spaced by a first comb spacing and shifted from zero by a firstCEO frequency, is locked to a first reference laser by controlling thefirst CEO frequency. In some embodiments, the first reference laser islocked to an atomic or molecular transition. In one example of step 708,LO tooth 406(1) is phase-locked to first reference laser 410(1) at thefirst frequency offset Δf_(A) ^((LO)) (see FIG. 4 ). In step 710, asecond tooth, of the first frequency comb, is locked to a secondreference laser by controlling the first comb spacing. In one example ofstep 710, LO tooth 406(2) is phase-locked to second reference laser410(2) at the second frequency offset Δf_(B) ^(LO)).

In step 712, a third tooth, of a second frequency comb of teethequally-spaced by a second comb spacing and shifted from zero by asecond CEO frequency, is locked to the first reference laser bycontrolling the second CEO frequency. In one example of step 712, SIGtooth 404(1) is phase-locked to first reference laser 410(1) at thethird frequency offset Δf_(A) ^((SIG)). In step 714, fourth tooth, ofthe second frequency comb, is locked to the second reference laser bycontrolling the second comb spacing. In one example of step 714, SIGtooth 406(2) is phase-locked to second reference laser 410(2) at thefourth frequency offset Δf_(B) ^((SIG)).

Method 700 also includes a step 716 to frequency-stabilize the secondreference laser to a RF frequency reference by controlling the secondreference laser to lock the first comb spacing. In one example of step716, first photodetector 512(1) and second laser stabilizer 534 of DFCS500 cooperate to measure the LO repetition rate f_(rep) ^((LO)) andcontrol second reference laser 410(2) to stabilize laser frequency f_(B)relative to the RF frequency reference. In one embodiment, step 716alternatively frequency-stabilizes the second reference laser to the RFfrequency reference by controlling the second reference laser to lockthe second comb spacing. In one example of step 716 of this embodiment,sixth photodetector 512(6) and second laser stabilizer 534 of DFCS 500cooperate to measure the SIG repetition rate f_(rep) ^((SIG)) andcontrol second reference laser 410(2) to stabilize laser frequency f_(B)relative to the RF frequency reference.

In some embodiments, method 700 includes a step 718 to spectrallybroadening the first and second frequency combs, in respective first andsecond nonlinear fibers, into first and second spectra that coverspectral lines of a gas to be spectroscopically detected with the firstand second spectra. In one example of step 718, first and secondnonlinear optical fiber 520(1), 520(2) of DFCS 500 spectrally broadenrespective amplified pulse trains 542, 552 such that respectivefrequency combs 120(1), 120(2) have a spectral width that coversspectral features (i.e., absorption lines) of interest in gas 116. Inone of these embodiments, the first and second spectra contain infraredlight. In another of these embodiments, the target gas is one or more ofmethane, carbon dioxide, carbon monoxide, hydrogen sulfide, acetylene,ethylene, propane, butane, ethane, and water vapor. In another of theseembodiments, method 700 includes spectroscopically measuring a gaseoussample with the first and second spectra to detect the target gas in thegaseous sample. In one example of spectroscopically measuring, pulsetrains 106, 108 traverse gas 116 to detect the target gas therein (seeFIG. 1 ).

In some embodiments, method 700 includes a step 702 to generate thefirst frequency comb with a first mode-locked laser, and a step 704 togenerate the second frequency comb with a second mode-locked laser. Inone of these embodiments, each of the first and second mode-lockedlasers is a femtosecond fiber laser. Each femtosecond fiber laser may anerbium-fiber laser operating in the infrared with a repetition ratef_(rep) between 10 and 1000 MHz. In one example of step 702, LO comblaser 510 generates first optical pulse train 540 that is subsequentlyprocessed (e.g., amplified and spectrally broadened) to become LOfrequency comb 120(1), and SIG comb laser 530 generates second opticalpulse train 550 that is subsequently processed to become SIG frequencycomb 120(2).

In embodiments that include the first and second mode-locked lasers, thefirst mode-locked laser may form a first cavity of a first cavitylength, wherein step 710 may be implemented by controlling the firstcavity length. Similarly, the second mode-locked laser may form a secondcavity of a second cavity length, wherein step 714 may be implemented bycontrolling the second cavity length. Furthermore, the first mode-lockedlaser may be powered by a first pump that outputs a first pump level(e.g., an optical power level), wherein step 708 may be implemented bycontrolling the first pump level. Similarly, the second mode-lockedlaser may be powered by a second pump that outputs a second pump level,wherein step 712 may be implemented by controlling the second pumplevel.

In some embodiments, method 700 includes a step 706 to lock the firstreference laser to an atomic or molecular transition such that thefrequency of the first reference laser is stabilized to the frequency ofthe atomic or molecular transition. In one example of step 706, firstlaser stabilizer 536 of DFCS 500 samples first coherent light 560 andcontrols first reference laser 410(1) to stabilize the frequency f_(A).First laser stabilizer 536 may include a vapor-cell spectrometer with avapor cell containing a gas that is spectroscopically probed by thesampled first coherent light 560 to generate a feedback signal forlocking f_(A) to a transition in the gas.

FIG. 7B is a flow chart of a frequency-stabilization method 752 forfield-deployable dual frequency-comb spectroscopy. Method 750 includessteps 756, 758, 760, 762, 764, and 766 that may occur in any order.Method 750 frequency-stabilizes teeth of first and second frequencycombs relative to a single RF frequency reference. Method 700 alsoestablishes coherence between the first and second frequency combs basedon coherence of first and second reference lasers. Method 700 may beimplemented with DFCS 600 of FIG. 6 .

Method 750 includes step 756 to lock a first CEO frequency, of a firstfrequency comb of teeth equally-spaced by a first comb spacing andshifted from zero by the first CEO frequency, to a RF frequencyreference by controlling the first CEO frequency. In one example of step756, f-2f interferometer 604 of FIG. 6 is used to detect the LO CEOfrequency f₀ ^((LO)), and LO comb stabilizer 642 phase-locks f₀ ^((LO))to a RF signal generated by a frequency synthesizer referenced to the RFfrequency reference.

Method 750 includes step 758 to lock a first tooth, of the firstfrequency comb, to a first reference laser by controlling the first combspacing. Method 750 includes step 760 to frequency-stabilize the firstreference laser to the RF frequency reference by controlling the firstreference laser to lock the first comb spacing. Steps 758 and 760cooperate to stabilize the first comb spacing. In one example of steps758 and 760, LO comb stabilizer 642 of FIG. 2 phase-locks the first beatΔf_(A) ^((LO)), to a RF signal generated by a frequency synthesizerreferenced to the RF frequency reference, by controlling LO comb laser510 to change the LO repetition rate f_(rep) ^((LO)). First laserstabilizer 636 then controls reference laser 410(1) to change the laserfrequency f_(A) such that the LO repetition rate f_(rep) ^((LO)), asmeasured by photodetector 512(1), is locked relative to the RF frequencyreference.

Method 750 includes step 762 to lock a second reference laser to asecond tooth of the first frequency comb. In one example of step 762,second laser stabilizer 634 of FIG. 6 controls the frequency f_(B) ofreference laser 410(2) to phase-lock the second beat Δf_(B) ^((LO)) to aRF signal generated by a frequency synthesizer referenced to the RFfrequency reference.

Method 750 includes step 764 to lock a third tooth, of a secondfrequency comb of teeth equally-spaced by a second comb spacing andshifted from zero by a second CEO frequency, to the first referencelaser by controlling the second comb spacing. Method 750 includes step766 to lock a fourth tooth, of the second frequency comb, to the secondreference laser by controlling the second CEO frequency. In one exampleof step 764, SIG comb stabilizer 532 of FIG. 6 phase locks the thirdbeat Δf_(A) ^((SIG)), to a RF signal generated by a frequencysynthesizer referenced to the RF frequency reference, by controlling SIGcomb laser 530 to change the SIG repetition rate f_(rep) ^((SIG)). Inone example of step 766, SIG comb stabilizer 532 phase locks the fourthbeat Δf_(B) ^((SIG)), to a RF signal generated by a frequencysynthesizer referenced to the RF frequency reference, by controlling SIGcomb laser 530 to change the SIG CEO frequency f₀ ^((SIG)).

In some embodiments, method 750 includes a step 752 to generate thefirst frequency comb with a first mode-locked laser, and a step 754 togenerate the second frequency comb with a second mode-locked laser. Inone of these embodiments, each of the first and second mode-lockedlasers is a femtosecond fiber laser. Each femtosecond fiber laser may anerbium-fiber laser operating in the infrared with a repetition ratef_(rep) between 10 and 1000 MHz. In one example of step 752, LO comblaser 510 generates first optical pulse train 540 that is subsequentlyprocessed (e.g., amplified and spectrally broadened) to become LOfrequency comb 120(1), and SIG comb laser 530 generates second opticalpulse train 550 that is subsequently processed to become SIG frequencycomb 120(2).

In embodiments that include the first and second mode-locked lasers, thefirst mode-locked laser may form a first cavity of a first cavitylength, wherein step 758 may be implemented by controlling the firstcavity length. Similarly, the second mode-locked laser may form a secondcavity of a second cavity length, wherein step 764 may be implemented bycontrolling the second cavity length. Furthermore, the first mode-lockedlaser may be powered by a first pump that outputs a first pump level(e.g., an optical power level), wherein step 756 may be implemented bycontrolling the first pump level. Similarly, the second mode-lockedlaser may be powered by a second pump that outputs a second pump level,wherein step 766 may be implemented by controlling the second pumplevel.

In some embodiments, method 750 includes spectrally broadening the firstand second frequency combs, in respective first and second nonlinearfibers, into first and second spectra that cover spectral lines of atarget gas to be spectroscopically detected with the first and secondspectra. In one example of spectrally broadening, first and secondnonlinear optical fiber 520(1), 520(2) of DFCS 600 spectrally broadenrespective amplified pulse trains 542, 552 such that respectivefrequency combs 120(1), 120(2) have a spectral width that coversspectral features (i.e., absorption lines) of interest of the targetgas. In one of these embodiments, the first and second spectra containinfrared light. In another of these embodiments, the target gas is oneor more of methane, carbon dioxide, carbon monoxide, hydrogen sulfide,acetylene, ethylene, propane, butane, ethane, and water vapor. Inanother of these embodiments, method 750 includes spectroscopicallymeasuring a gaseous sample with the first and second spectra to detectthe target gas in the gaseous sample. In one example ofspectroscopically measuring, pulse trains 106, 108 traverse gas 116 todetect the target gas therein (see FIG. 1 ).

Embodiments with More than Two Optical Frequency Combs

In DFCS 500 and DFCS 600, first optical frequency comb 120(1), asgenerated by LO comb laser 510, is used for both stabilizing referencelasers 410 and outputting LO pulse train 210(1) to form double pulsetrain for DCS. In embodiments, reference-laser stabilization and LOpulse-train formation for DCS are implemented with two optical frequencycombs generated by separate comb lasers, thereby increasing the overallnumber of optical frequency combs from two to three. Similar to DFCS 500and DFCS 600, embodiments with three optical frequency combs can beadvantageously implemented with at most one f-2f interferometer or onevapor-cell spectrometer for locking to an atomic or moleculartransition.

In some embodiments of a DFCS with three optical frequency combs, DFCS500 of FIG. 5 is expanded to include a third optical frequency comb thatis generated by a third comb laser whose frequency components, orthird-comb teeth, are stabilized using any technique, or combinations oftechniques, known in the art. For example, the third comb may span anoctave and include an f-2f interferometer (e.g., see f-2f interferometer604 of FIG. 6 ) to detect and phase-lock a third-comb CEO frequency to aRF frequency reference. A repetition rate of the third comb may bephase-locked to the RF frequency reference by directly controlling thecavity spacing of the third comb laser, or via control of a referencelaser to which the third comb is locked (i.e., similarly to how secondlaser stabilizer 534 of FIG. 5 phase-locks the LO repetition ratef_(rep) ^((LO)) by controlling the frequency f_(B) of second referencelaser 410(2)). In another example, one of the third-comb CEO frequencyand the third-comb repetition rate is frequency-stabilized byphase-locking one third-comb tooth to a stabilized reference laser thatis different from first and second reference lasers 410(1), 410(2). Inanother example, both the third-comb CEO frequency and the third-combrepetition rate are frequency-stabilized by phase-locking two third-combteeth to two corresponding stabilized reference lasers different fromfirst and second reference lasers 410(1), 410(2). Each of the stabilizedreference lasers may be a laser whose frequency is locked to an atomicor molecular transition (e.g., via Doppler-free spectroscopy of a gas ina vapor cell).

The frequency stability of the third-comb teeth is transferred to firstand second reference lasers 410(1), 410(2) by phase-locking first andsecond reference lasers 410(1), 410(2) to two different third-combteeth. The phase-locking of first reference laser 410(1) can beimplemented with first laser stabilizer 536 using a heterodyne beatbetween first reference laser 410(1) and a first third-comb tooth.Similarly, the phase-locking of second reference laser 410(2) can beimplemented with second laser stabilizer 534 using a heterodyne beatbetween second reference laser 410(2) and a second third-comb tooth(rather than using the LO repetition rate f_(rep) ^((LO)), as shown inFIG. 5 ).

The frequency stability of the third-comb teeth is then transferred toLO frequency comb 120(1) by phase-locking two of LO teeth 404 to firstand second reference lasers 410(1), 410(2) in the same manner describedabove for FIG. 5 . The frequency stability of the third-comb teeth isalso transferred to SIG frequency comb 120(2) by phase-locking two ofSIG teeth 406 to first and second reference lasers 410(1), 410(2) in thesame manner described above for FIG. 5 .

In some embodiments, a three-frequency-comb DFCS uses a frequencymicrocomb for the third optical frequency comb. Advantageously, afrequency microcomb (also called a chip-based frequency comb or amicroresonator frequency comb) is significantly smaller than aconventional frequency comb (e.g., one based on a Ti:Saph or fiberfemtosecond laser) and therefore can help facilitate portability androbustness of a DCS spectrometer. However, due to its small cavitylength, a frequency microcomb typically has a repetition rate so highthat the pulse-train outputted by the microcomb cannot be used for DCS.Nevertheless, a frequency microcomb may still be used as an aide forstabilizing reference lasers for DCS, as described above.

In some embodiments, more than two optical frequency combs (i.e., LOcomb laser 510 and SIG comb laser 530) are frequency-stabilized to thethird frequency comb. For example, fourth and fifth frequency combs canbe phase-locked to first and second reference lasers 410(1), 410(2)similarly to first and second optical frequency combs 120(1) and 120(2).Pulse trains outputted by the fourth and fifth frequency combs can becombined to generate a second double pulse train for DCS. The seconddouble pulse train may be configured differently from double pulse train106 (e.g., different repetition rates and spectral broadening) such thatboth double pulse trains simultaneously probe a gas sample withdifferent spectra (e.g., to detect different target species in the sameor different spectral features of the same target species). Additionalfrequency combs can be similarly locked to first and second referencelasers 410(1), 410(2) without departing from the scope hereof.

In some embodiments, more than two reference lasers are stabilized tothe third frequency comb. For example, third and fourth reference laserscan be phase-locked to the third optical frequency comb similarly tofirst and second reference lasers 410(1), 410(2). In embodiments wherefourth and fifth frequency combs are used (e.g., to generate a seconddouble pulse train, as described above), the fourth and fifth frequencycombs may be phase-locked to the third and fourth reference lasersinstead of first and second reference 410(1), 410(2). The third andfourth reference lasers may operate at different wavelengths than firstand second reference lasers 410(1), 410(2). The third and fourthreference lasers may be selected such that their wavelengths betteroverlap the spectra of the fourth and fifth frequency combs, as comparedto the wavelengths of first and second reference lasers 410(1), 410(2).The improved spectral overlap increases the SNR of heterodyne beatsdetected between each of the third and fourth reference lasers and eachof the fourth and fifth frequency combs, advantageously improvingrobustness of the phase-lock loops used to frequency stabilize thefourth and fifth frequency combs.

DCS Stabilization Based on Walking Interferograms

As described above for FIGS. 5 and 6 , an optical frequency comb can becalibrated using a single-frequency reference laser (e.g., firstreference laser 410(1)) whose frequency is initially known to within thecomb spacing f_(rep) (e.g., typically between 50 and 1000 MHz). Initialknowledge of the laser frequency to this level of accuracy uniquelydetermines the integer n that indexes the comb tooth closest to thelaser frequency. An initial value for the laser frequency may beobtained with a high-resolution wavemeter. However, such a wavemeterincreases the size, weight, and complexity of DCS system 100, and thusinhibits portability of DCS system 100. Alternatively, the laserfrequency can be determined by locking the reference laser to an atomicor molecular transition with a known transition frequency. However, thereference laser may have a frequency that is spectrally far from anyconvenient transition that could be used for such locking.

FIGS. 8 and 9 illustrate a frequency measurement method that utilizes adual frequency-comb source as an optical frequency meter to measure thelaser frequency f_(A) of first reference laser 410(1). With this method,the frequency f_(A) can be advantageously measured, relative to a RFfrequency reference, without a high-resolution wavemeter and withoutrelying on spectroscopic knowledge of an atomic or molecular transition.Thus, the frequency-measurement method advantageously eliminates theneed for a dedicated high-performance wavemeter, enabling portability byreducing size, power consumption, and component count.

The frequency-measurement method requires the two frequency combs of thedual frequency-comb source to have their CEO frequencies locked relativeto the RF frequency reference. Thus, each of the two frequency combsoperates with an f-2f interferometer (e.g., f-2f interferometer 604 ofFIG. 6 ). Since many prior-art DCS systems already operate with f-2finterferometers to stabilize the CEO frequencies, thefrequency-measurement method embodiments presented herein can implementnew functionality with existing DCS systems.

For clarity in the following discussion, the frequency-measurementmethod is described for measuring the frequency f_(A) of first referencelaser 410(1). More specifically, since it is assumed that the CEOfrequencies of both frequency combs are detected and locked, secondreference laser 410(2) is not needed to fully stabilize the twofrequency combs. However, it should be appreciated that thefrequency-measurement method can be similarly applied to anysingle-frequency laser to which the two frequency combs are locked.

First reference laser 410(1) may be free-running or locked. For example,first reference laser 410(1) may be locked to a resonance, of aFabry-Perot cavity, with an unknown resonant frequency, wherein thefrequency-measurement method can be used to measure the resonantfrequency. Alternatively, first reference laser 410(1) may be locked toan atomic or molecular transition with an unknown transition frequency,wherein the frequency-measurement method can be used to measure thetransition frequency.

FIG. 8 shows a temporal sequence 800 of interferograms 302 in which a“walking” center burst 306 is used to determine the laser frequencyf_(A) of first reference laser 410(1). Both frequency combs 120(1) and120(2) are locked to reference laser 410(1), at respective offsetfrequencies Δf_(A) ^((SIG)) and Δf_(A) ^((LO)), as shown in FIG. 4 . Asdescribed in more detail below, the temporal evolution of center burst306 in temporal sequence 800 can be used to determine the number ofteeth in each of a plurality of Nyquist windows (see Nyquist windows 902in FIG. 9 ) from which the laser frequency f_(A) can be determined.

FIG. 9 is a frequency-domain plot showing the laser frequency f_(A) ofreference laser 410(1) relative to LO and SIG teeth 406, 404 of LO andSIG frequency combs 120(1), 120(2), respectively. Frequency f_(A) can beexpressed mathematically as

f _(A) =f ₀ ^((LO)) +n _(A) ^((LO)) f _(rep) ^((LO)) +Δf _(A) ^((LO)) =f₀ ^((SIG)) +n _(A) ^((SIG)) f _(rep) ^((SIG)) +Δf _(A) ^((SIG)).   (1)

For simplicity, it is assumed that f₀ ^((LO))=f₀ ^((SIG))=0, in whichcase the zeroth-mode frequency components of both frequency combs120(1), 120(2) coincide exactly at 0 Hz, as shown in FIG. 9 . It is alsoassumed that Δf_(A) ^((LO))=Δf_(A) ^((SIG))=0 such that f_(A) has thesame frequency as one LO tooth 406(1) and one SIG tooth 404(1). Withthese assumptions, Eqn. 1 simplifies to

f _(A) =n _(A) ^((LO)) f _(rep) ^((LO)) =n _(A) ^((SIG)) f _(rep)^((SIG)).  (2)

Repetition rate f_(rep) ^((LO)) and/or f_(rep) ^((SIG)) may becontrolled to form Nyquist windows 902, each with a frequency spandf_(Ny). At integer multiples of df_(Ny), LO teeth 406 and SIG teeth 404coincide to form upper and lower anchor frequencies for Nyquist windows902. In each of Nyquist windows 902, a number n_(q)+1 of LO teeth 406 isone greater than a number n_(q) of SIG teeth 404. The number n_(q) ofteeth in one Nyquist window is defined herein to be inclusive of onlyone of the two anchor frequencies of the Nyquist window. With thisdefinition, n_(q) equals the number of comb spacings (i.e., f_(rep)^((LO)) or f_(rep) ^((SIG))) between the lower and upper anchorfrequencies of the Nyquist window. Thus, n_(q)=9 in the example of FIG.9 , wherein each of Nyquist windows 902 has ten LO teeth 406corresponding to a frequency span of 10f_(rep) ^((LO)), and nine SIGteeth 404 corresponding to a frequency span of 9f_(rep) ^((SIG)). WhileFIG. 9 shows n_(q) as an integer, in general n_(q) does not need to bean integer. Each of Nyquist windows 902 may have any number of teeth(e.g., hundreds of thousands) without departing from the scope hereof.

Based on the above definition of n_(q), the frequency span df_(Ny) ofeach of Nyquist windows 902 is given by

df _(Ny)=(n _(q)+1)f _(rep) ^((LO)) =n _(q) f _(rep) ^((SIG)) =n _(q)(f_(rep) ^((LO)) +Δf _(rep)),  (3)

where Δf_(rep)=f_(rep) ^((SIG))−f_(rep) ^((LO)) is a difference inrepetition rates of LO and SIG frequency combs 120(1), 120(2). From Eqn.3,

$\begin{matrix}{{n_{q} = {\frac{f_{rep}^{({LO})}}{\Delta f_{rep}} = {\frac{f_{rep}^{({LO})}}{f_{rep}^{({SIG})} - f_{rep}^{({LO})}} = \frac{1}{{f_{rep}^{({SIG})}/f_{rep}^{({LO})}} - 1}}}},} & (4)\end{matrix}$

which expresses n_(q) in terms of f_(rep) ^((SIG)) and f_(rep) ^((LO)).Combining Eqns. 3 and 4 yields

df _(Ny) =f _(rep) ^((SIG)) f _(rep) ^((LO)) /Δf _(rep),  (5)

which expresses the frequency span df_(Ny) of each of Nyquist windows902 in terms of the repetition rates f_(rep) ^((SIG)) and f_(rep)^((LO)).

Repetition rate f_(rep) ^((LO)) and/or f_(rep) ^((SIG)) may be furthercontrolled such that the boundary of one of Nyquist windows 902 occursat frequency f_(A). For example, in FIG. 9 frequency f_(A) coincideswith the third multiple of df_(Ny), which is the lower anchor frequencyof Nyquist window 902(4) and the upper anchor frequency of Nyquistwindow 902(3). The frequency of LO tooth 406(1), which is indexed byinteger n_(A) ^((LO)) in Eqn. 2, is 3df_(Ny). Similarly, the frequencyof SIG tooth 404(1), which is indexed by integer n_(A) ^((SIG)) in Eqn.2, is also 3df_(Ny). Therefore, SIG mode n_(A) ^((SIG)) can be expressed

n _(A) ^((SIG)) =N _(Ny) n _(q),  (6)

where N_(Ny) is a “Nyquist number” that indexes Nyquist windows 902.Eqn. 2 may be expressed in terms of N_(Ny) as

f _(A) =N _(Ny)(n _(q)+1)f _(rep) ^((LO)) =N _(Ny) n _(q) f _(rep)^((SIG)).  (7)

Eqn. 7 may be used to determine f_(A) when unknown, such as when DCSsystem 100 is initially powered on (i.e., a “cold start”), or when f_(A)changes frequency (e.g., drift). To use Eqn. 7, three pieces ofinformation are needed: (1) a measurement of either f_(rep) ^((SIG)) orf_(rep) ^((LO)), (2) the Nyquist number N_(Ny), and (3) the number n_(q)of comb teeth in each of Nyquist windows 902.

The first piece of information needed to determine f_(A) with Eqn. 7 isa measurement of either f_(rep) ^((LO)) or f_(rep) ^((SIG)). As shownbelow, N_(Ny) and n_(q) are integers that can each be determineduniquely, and therefore do not contribute to any uncertainty in theresulting value of f_(A). Rather, the uncertainty in f_(A) is determinedsolely by the measurement of f_(rep). For most trace-gas spectroscopyapplications, f_(A) only needs to be known to a relative accuracy of10⁻⁷ to properly identify spectral lines of the trace gas; f_(rep) canbe readily measured to this level of accuracy using a compact,low-resolution frequency counter. It is assumed that f_(A) is stable atthe 10⁻⁷ level, or better, so that fluctuations in f_(A) do notcontribute to the uncertainty of the measurement of f_(rep).

The second piece of information needed to determine f_(A) with Eqn. 7 isthe Nyquist number N_(Ny) of the one of Nyquist windows 902 in whichf_(A) lies. When f_(A) coincides with anchor frequencies, it is assumedthat f_(A) lies in the Nyquist window 902 with the lower Nyquist number.Thus, in FIG. 9 , f_(A) coincides with the upper anchor frequency ofNyquist window 902(3), and thus f_(A) lies in Nyquist window 902(3) forwhich N_(Ny)=3. More generally, the Nyquist number can be determinedfrom

$\begin{matrix}{{N_{Ny} = {\lbrack \frac{f^{(e)}}{{df}_{Ny}} \rbrack = \lbrack \frac{f^{(e)}\Delta f_{rep}}{f_{rep}^{({SIG})}f_{rep}^{({LO})}} \rbrack}},} & (8)\end{matrix}$

where the brackets [x] indicate the nearest integer to x, and f^((e)) isa low-resolution estimate for f_(A). Typical values for df_(Ny) arebetween 5 and 10 THz, while an infrared or visible laser operates at afrequency between 100-1000 THz. Thus, N_(Ny) is no larger than 200. Toidentify N_(Ny) uniquely to the nearest integer, each of the variablesin Eqn. 8 only needs to be measured to the 0.5% level, or better. Thenominal operating wavelength of the laser (e.g., as specified by themanufacturer, or measured with a low-resolution wavemeter) can be usedto obtain f^((e)) to better than this level. Each of f_(rep) ^((SIG)),f_(rep) ^((LO)), and Δf_(rep) can be directly measured to better thanthis level using a low-resolution frequency counter operating with afast gate time (e.g., less than 1 s).

The third piece of information needed to determine f_(A) with Eqn. 7 isthe number n_(q) of comb teeth in each of Nyquist windows 902. FIG. 8shows how temporal sequence 800 can be used to quickly and uniquelydetermine n_(q) using the same detectors and signal-processing equipmentalready needed to spectroscopically measure gas 116 with double pulsetrain 106. In FIG. 8 , the repetition rates f_(rep) ^((SIG)) and f_(rep)^((LO)) have been selected such that center burst 306 “walks” by anamount Δt from one interferogram 306 to the next. More specifically,center burst 306 is shown in FIG. 8 as a first center burst 306(1) thatoccurs at time t=0 in a first interferogram 302(1), a second centerburst 306(2) that is temporally shifted by Δt in a subsequent secondinterferogram 302(2), a third center burst 306(3) that is temporallyshifted by 2Δt in a subsequent third interferogram 302(3), and so on.This walking, or accumulation of temporal shifts Δt, continues until aninteger number u of shifts Δt equals the frame time T_(f), at whichpoint center burst 306 again occurs at time t=0. In the example of FIG.8 , u=5, wherein a fifth center burst 306(5) occurs at the beginning ofa sixth interferogram 302(6). Temporal sequence 800 of u interferograms302 continuously repeats, starting with a (u+1)th interferogram andevery u^(th) interferogram thereafter. Thus, in FIG. 8 , temporalsequence 800 begins again at an eleventh interferogram, a sixteenthinterferogram, a twenty-first interferogram, etc.

Temporal shift Δt may be alternatively expressed as u=n_(i)/|R|, wheren_(i) is the known number of equally-spaced data points in eachinterferogram 302, and R is a “walking” rate equal to the number of datapoints in temporal shift Δt. Each data point is acquired in a time1/f_(rep) ^((LO)), and thus the frame time T_(f)=n_(i)/f_(rep) ^((LO))is the total time needed to acquire all n_(i) points in oneinterferogram 302. Furthermore, the frame time T_(f) may be related to acenter-burst time T_(cb)=1/Δf_(rep) (see FIG. 8 ) that equals the timebetween sequential center bursts 306:

uT _(f)=(u±1)T _(cb).  (9)

where the plus sign corresponds to T_(cb)>T_(f). (i.e., center burst 306“walks” to the right) and the negative sign corresponds to T_(cb)<T_(f).(i.e., center burst 306 “walks” to the left). Combining Eqns. 4 and 9yields:

$\begin{matrix}{{n_{q} = {\frac{f_{rep}^{({LO})}}{\Delta f_{rep}} = {\frac{n_{i}^{2}}{n_{i} \pm {❘R❘}} \approx {n_{i} + R}}}},} & (10)\end{matrix}$

where the approximation assumes R«n_(i). Since n_(i) is known from aselected digitization rate (i.e., time step 314 in FIG. 3 ), n_(q) maybe found using one of several ways to measure the walking rate R. Forexample, when center burst 306 is relatively narrow, a simplecenter-burst tracker can be implemented to obtain a center position ofcenter burst 306 from one interferogram 302 by identifying the one datapoint of said one interferogram 302 at which center burst 306 has amaximal value. A change in the center position can be measured betweenconsecutive interferograms 302 to obtain the walking rate R.

The simple center-burst tracker can only detect a shift of the centerposition that equals an integer number of data points (i.e., an integermultiple of time step 314 in FIG. 3 ). However, n_(q) need not be aninteger. Specifically, rearranging Eqn. 6 gives n_(q)=n_(A)^((SIG))/N_(Ny), where n_(A) ^((SIG)) and N_(Ny) are both integers.Thus, in the most general case, n_(q) may be a non-integer rationalnumber. In the limit R«n_(i), Eqns. 4 and 8 may be combined to obtain

$\begin{matrix}{{R \approx {n_{q} - n_{i}}} = {\frac{n_{A}^{({SIG})} + {n_{i}N_{Ny}}}{N_{Ny}}.}} & (11)\end{matrix}$

Thus, when n_(q) is a non-integer rational number, then so is R. In thiscase, the center position of center burst 306 will shift by an integernumber of data points after N_(Ny) consecutive interferograms 302. Thesimple center-burst tracker described above may be additionallyconfigured to measure a shift of the center position across N_(Ny)consecutive interferograms 302, such that the measured shift correspondsto an integer number of data points. The simple center-burst tracker maythen divide the measured shift by N_(Ny) to obtain the non-integerrational value for R, and thus n_(q).

Phase noise and/or a chirped pulse will broaden center burst 306,reducing the SNR of interferograms 302, thereby increasing theuncertainty of the determined location of center burst 306 (i.e., thelocation of the maximal data point). In such a situation, severaladditional interferograms 302 can be used to increase the SNR andthereby measure R with an uncertainty low enough that n_(q) can beuniquely determined. To increase the SNR, an advanced center-bursttracker may be used to obtain a center position of center burst 306 fromone interferogram 302 by monitoring a phase walk of the detectedfrequency spectrum, or by fitting an envelope function to center burst306, wherein the center position is a fit parameter of the envelopefunction. However, the SNR may also be increased by averaging severalmeasurements of R obtained with the simple center-burst trackerdescribed above. This latter solution is usually adequate given howrapidly measurements of R can be acquired (i.e., typically oneinterferogram 302 every 1/Δf_(rep)≈1 ms). In addition, continuousaveraging of measurements of R is enabled by the fact that n_(q) dependsonly on n_(A). Thus, once frequency combs 120(1), 120(2) are locked toreference laser 410(1), barring phase slips, both n_(q) and R will beconstant, even if f_(A) drifts.

An alternative way to measure n uses Eqn. 4 with measurements of f_(rep)^((LO)) and Δf_(rep) (or, alternatively, f_(rep) ^((SIG)) and Δf_(rep)).For repetition rates of approximately 100 MHz, and a Nyquist frequencyspan df_(Ny) of 10 THz, n_(q)≈10⁵. Thus, to obtain a unique value forn_(q) with this approach, f_(rep) ^((LO)) and Δf_(rep) should each bemeasured at the 10⁻⁶ level, or better. Repetition rate f_(rep) ^((LO))(or, alternatively, f_(rep) ^((SIG))) can be measured to better thanthis level using a low-resolution frequency counter operating with fastgate times (e.g., less than 1 s). Repetition rates f_(rep) ^((LO)) andf_(rep) ^((SIG)) may be combined in a mixer to form Δf_(rep), which canalso be directly measured with a low-resolution frequency counter. Themeasured value of Δf_(rep) may then be divided into the measured valueof f_(rep) ^((LO)) to obtain n_(q). When f_(A) has a lot of jitter,f_(rep) ^((LO)) and Δf_(rep) may be measured simultaneously using twosynchronized frequency counters.

It should be understood by those trained in the art that sinceΔf_(rep)«f_(rep) ^((LO)),f_(rep) ^((SIG)), calculating Δf_(rep) bysubtracting a value of f_(rep) ^((LO)) from a value of f_(rep) ^((SIG))leads to a significant loss in precision. For example, if the repetitionrates are each approximately 100 MHz, and Δf_(rep)˜100 Hz, then the sixmost significant digits of f_(rep) ^((LO)) and f_(rep) ^((SIG)) cancelupon subtraction, reducing the precision of Δf_(rep) by 10⁶. To overcomethis loss of precision f_(rep) ^((LO)) and f_(rep) ^((SIG)) would eachneed to be measured at the 10⁻¹² level, or better. Measuring therepetition rates at this resolution requires high-performance,high-stability frequency counters whose size and power consumptioninhibits portability and field-deployment of DCS system 100. It alsorequires long gate times (e.g., 100 seconds, or more) during whichf_(rep) ^((LO)) and f_(rep) ^((SIG)) can drift. For these reasons, it ispreferable to generate Δf_(rep) by combining f_(rep) ^((LO)) and f_(rep)^((SIG)) in a mixer, as described above, and directly counting Δf_(rep).with a conventional, low-resolution frequency counter.

Another way to measure n_(q) uses Eqn. 4 with a measurement of thefrequency ratio f_(rep) ^((SIG))/f_(rep) ^((LO)). This ratio can bemeasured to better than 10⁻⁶ using a low-resolution frequency counterconfigured with two inputs, one that receives f_(rep) ^((SIG)) and onethat receives f_(rep) ^((LO)). The measured value of the ratio may beused to calculate n_(q) via Eqn. 5.

In the preceding discussion, it was assumed that f₀ ^((LO))=f₀^((SIG))=Δf_(A) ^((LO))=Δf_(A) ^((SIG))=0, which simplified FIG. 9 byaligning LO teeth 406 and SIG teeth 404 at the boundaries of Nyquistwindows 902 (i.e., at 0 Hz and integer multiples of df_(Ny)). However,the preceding arguments remain valid for arbitrary values of f₀ ^((LO)),f₀ ^((SIG)), Δf_(A) ^((LO)), and f_(A) ^((SIG)) provided that f_(A) isnear the edge of a Nyquist window; this can be shown by rederiving Eqn.7 using Eqn. 1 rather than Eqn. 2.

FIG. 10 is a flow chart of a method 1000 for measuring the frequency ofa laser with a dual frequency-comb spectrometer. Method 1000 includes astep 1006 to measure a walking rate of a plurality of center bursts in asequence of consecutive interferograms recorded with the dualfrequency-comb spectrometer. In one example of step 1006, a centerposition of center bursts 306 is tracked over time to determine thewalking rate R. In one embodiment, the center position is tracked byfitting each center burst to an envelope function to obtain a fittedcenter said each center burst, and determining the shift from the fittedcenters. This embodiment may be implemented with the advancedcenter-burst tracker described above.

Method 1000 also includes a step 1008 to determine, from the measuredwalking rate and a number of data points in each of the interferograms,a number of teeth in each of a plurality of Nyquist windows formed bythe dual frequency-comb spectrometer. In one example of step 1008, thenumber n_(q) of SIG teeth 406 in each of Nyquist windows 902 isdetermined using Eqn. 10. Method 1000 also includes a step 1010 todetermine a Nyquist number of one of the Nyquist windows that covers thefrequency of the laser. In one example of step 1010, the Nyquist numberis calculated using Eqn. 8. Method 1000 also includes a step 1012 todetermine the frequency of the laser from (i) the number of teeth, (ii)the Nyquist number, and (iii) a measurement of a comb spacing of one offirst and second frequency combs of the dual frequency-combspectrometer. In one example of step 1012, the frequency f_(A) of firstreference laser 410(1) is calculated using Eqn. 7. In one embodiment,method 1000 includes a step 1014 to output the frequency of the laser tocalibrate the dual frequency-comb spectrometer for spectroscopicallydetecting a gas with the dual frequency-comb spectrometer.

In some embodiments, method 1000 includes a step 1002 to lock a firsttooth, of a first plurality of teeth of the first frequency comb, to thelaser at a first offset frequency, and a step 1004 to lock a secondtooth, of a second plurality of teeth of the second frequency comb, tothe laser at a second offset frequency. In one example of steps 1002 and1004, LO tooth 406(1) is phase-locked to reference laser 410(1) at thefirst frequency offset Δf_(A) ^((LO)) (see FIG. 4 ), and SIG tooth404(1) is phase-locked to reference laser 410(1) at the third frequencyoffset Δf_(A) ^((SIG)). The first tooth, second tooth, first offsetfrequency, and second offset frequency may be selected so that the firstand second frequency combs form the Nyquist windows.

Locking Conditions for Immunity to RF Reference Frequency Drift

To form a Nyquist window for detecting gas 116, first and secondfrequency combs 120(1), 120(2) must be locked with appropriate choicesof the first, second, third, and fourth frequency offsets Δf_(A)^((LO)), Δf_(B) ^((LO)), Δf_(A) ^((SIG)), and Δf_(B) ^((SIG)). With theselected values of the four frequency offsets, drift of the referencefrequencies f_(A) and/or f_(B) may cause the center burst to walk fromone interferogram to the next. Furthermore, the walking rate of thecenter burst will change as the frequencies f_(A) and f_(B) drift. Witha walking center burst, multiple interferograms cannot be averagedtogether to improve the SNR (see FIG. 14A and the accompanyingdescription below).

One way to prevent walking is to select the frequency offsets to beequal, i.e., Δf_(A) ^((LO))=Δf_(B) ^((LO))=Δf_(A) ^((SIG))=Δf_(B)^((SIG)). However, this restricts the location of the Nyquist window infrequency space, i.e., the lower and upper anchor frequencies thatdefine the boundaries of the Nyquist window. For example, when Δf_(A)^((LO))=Δf_(A) ^((SIG)), LO tooth 406(1) and SIG tooth 404(1) coincideto establish an anchor frequency near reference frequency f_(A) (i.e.,within ±f_(rep)/2). Similarly, when Δf_(B) ^((LO))=Δf_(B) ^((SIG)), LOtooth 406(2) and SIG tooth 404(2) coincide to establish another anchorfrequency near reference frequency f_(B). In some applications, it ispreferable to establish anchor frequencies farther from the referencefrequencies (e.g., by several terahertz).

Presented in this section are drift-immune DCS frequency-stabilizationtechniques that ensure that center bursts do not walk (i.e., arestationary), and thus can be advantageously averaged together to improvethe SNR. These “no-walking” embodiments loosen the restrictions on thefrequency offsets Δf_(A) ^((LO)), Δf_(B) ^((LO)), Δf_(A) ^((SIG)),Δf_(B) ^((SIG)), as compared to the above example where they are allequal, and thus facilitate the implementation of Nyquist windows havingany desired location and width (i.e., the difference between upper andlower anchor frequencies). Advantageously, no-walking embodiments keepthe center bursts stationary in the presence of certain types offrequency drift, particularly frequency drift of a common RF frequencyreference from which the frequency offsets are synthesized. With thisreduced sensitivity to frequency drift, an oven-controlled crystaloscillator (OCXO), for example, may be used as the RF frequencyreference in lieu of a larger, higher-power atomic frequency standard.

One aspect of the no-walking embodiments is the realization that centerbursts do not walk when

Δf _(A) ^((SIG)) +Δf _(B) ^((LO)) =Δf _(A) ^((LO)) +Δf _(B)^((SIG)).  (12)

Eqn. 12 is also referred to herein as the “no-walking condition”. Toderive Eqn. 12, consider first and second frequency combs 120(1), 120(2)locked to reference frequencies f_(A) and f_(B), as shown in FIG. 4 .The frequency f_(A) may be expressed in terms of parameters of firstfrequency comb 120(1), and in terms of parameters of second frequencycomb 120(2):

$\begin{matrix}\begin{matrix}{f_{A} = {f_{0}^{({LO})} + {n_{A}^{({LO})}f_{rep}^{({LO})}} + f_{A}^{({LO})}}} \\{= {f_{0}^{({SIG})} + {n_{A}^{({SIG})}f_{rep}^{({SIG})}} + {f_{A}^{({SIG})}.}}}\end{matrix} & (13)\end{matrix}$

The reference laser frequency f_(B) may be similarly expressed in termsof parameters of frequency combs 120(1), 120(2):

$\begin{matrix}\begin{matrix}{f_{B} = {f_{0}^{({LO})} + {n_{B}^{({LO})}f_{rep}^{({LO})}} + f_{B}^{({LO})}}} \\{= {f_{0}^{({SIG})} + {n_{B}^{({SIG})}f_{rep}^{({SIG})}} + {f_{B}^{({SIG})}.}}}\end{matrix} & (14)\end{matrix}$

Eqns. 13 and 14 may be combined to solve for the repetition ratesf_(rep) ^((SIG)) and f_(rep) ^((LO)).

$\begin{matrix}{{f_{rep}^{({LO})} = \frac{f_{A} - f_{B} - {\Delta f_{A}^{({LO})}} + {\Delta f_{B}^{({LO})}}}{n_{A}^{({LO})} - n_{B}^{({LO})}}};} & (15)\end{matrix}$$f_{rep}^{({SIG})} = {\frac{f_{A} - f_{B} - {\Delta f_{A}^{({SIG})}} - {\Delta f_{B}^{({SIG})}}}{n_{A}^{({SIG})} - n_{B}^{({SIG})}}.}$

The number of SIG comb teeth 404 in a Nyquist window is given byn_(q)=f_(rep) ^((LO))/(f_(rep) ^((SIG))−f_(rep) ^((LO))) (see Eqn. 4),wherein the number of LO comb teeth 406 in the Nyquist window isn_(q)+1. Walking of the center burst occurs when n_(q) does not equal aninteger frame length of the interferogram; the frame length is thenumber of discrete data points in one interferogram (e.g., the number ofsampled data points 310 of FIG. 3 ). As described previously, the numbern_(q) of teeth in one Nyquist window is defined herein to be inclusiveof only one of the two anchor frequencies of the Nyquist window so thatn_(q) equals the number of comb spacings (i.e., f_(rep) ^((LO)) orf_(rep) ^((SIG))) between the lower and upper anchor frequencies of theNyquist window.

The expression for n_(q) may be simplified by defining df≡f_(B)−f_(A),Δn^((LO))≡n_(B) ^((LO))−n_(A) ^((LO)) and Δn^((SIG))≡n_(B)^((SIG))−n_(A) ^((SIG)). In addition, the frequency offsets may bewritten as a fraction of a common RF reference frequency f_(c), i.e.Δf_(A) ^((LO))=α^((LO))f_(c), Δf_(B) ^((LO))=β^((LO))f_(c), Δf_(A)^((SIG))=α^((SIG))f_(c), and Δf_(B) ^((SIG))=β^((SIG))f_(c) forconstants α^((LO)), β^((LO)), α^((SIG)), and β^((SIG)). The result is:

$\begin{matrix}{{n_{q} = {\frac{f_{rep}^{({LO})}}{f_{rep}^{({SIG})} - f_{rep}^{({LO})}} = {- \frac{\Delta{n^{({LO})}( {{df} + \alpha^{({LO})} - \beta^{({LO})}} )}}{K}}}};} & (16)\end{matrix}$

where the denominator K is given by

K=df(Δn ^((LO)) −Δn ^((SIG)))+f _(c)(Δn ^((LO))(α^((SIG))+β^((SIG)))+Δn^((SIG))(α^((LO))+β^((LO)))).  (17)

Consider changes to n_(q) arising from changes (i.e., drift) of the RFreference frequency f_(c) and reference frequency difference Δf:

$\begin{matrix}{{\frac{\partial n_{q}}{\partial f_{c}} = \frac{{df}\Delta n^{({LO})}\Delta{n^{({SIG})}( {\alpha^{({LO})} - \alpha^{({SIG})} + \beta^{({SIG})} - \beta^{({LO})}} )}}{K}},} & (18)\end{matrix}$$\frac{\partial n_{q}}{{\partial\Delta}f} = {\frac{f_{c}\Delta n^{({LO})}\Delta{n^{({SIG})}( {{- \alpha^{({LO})}} + \alpha^{({SIG})} - \beta^{({SIG})} + \beta^{({LO})}} )}}{K}.}$

In both cases, when the constraint

−α^((LO))+α^((SIG))−β^((SIG))+β^((LO))=0  (19)

is satisfied, ∂n_(q)/∂f_(C)=0 and ∂n_(q)/∂Δf=0. Multiplying Eqn. 19 byf_(c) leads to the no-walking condition of Eqn. 12. Thus, when the fouroffset frequencies Δf_(A) ^((SIG)), Δf_(A) ^((LO)), Δf_(B) ^((SIG)),Δf_(B) ^((LO)) are selected to both satisfy Eqn. 12 and establish aNyquist window, changes in n_(q) due to changes in f_(c) and Δf aresuppressed, and thus stationary interferograms remain stationaryregardless of drift of f_(c) and Δf.

LO comb stabilizer 542 and SIG comb stabilizer 532 of FIG. 5 may beconfigured to implement the no-walking condition of Eqn. 12 with DFCS500, thereby creating one embodiment of a drift-immune dualfrequency-comb spectrometer. Similarly, LO comb stabilizer 642 of FIG. 6may cooperate with SIG comb stabilizer 532 to implement the no-walkingcondition with DFCS 600, thereby creating another embodiment of adrift-immune dual frequency-comb spectrometer. In these embodiments, thefrequency f_(c) is the output of the common RF frequency referencedescribed previously with respect to FIGS. 5 and 6 .

FIG. 11 is a flow chart of a drift-immune frequency-stabilization method1100 for locking a dual frequency-comb spectrometer having first andsecond frequency combs. Method 1100 frequency-stabilizes teeth of thefirst and second frequency combs to first and second lasers according tothe no-walking condition of Eqn. 12. Method 1100 may be implemented withany dual frequency-comb spectrometer that is locked to two referencelasers, including DFCS 500 of FIG. 5 and DFCS 600 of FIG. 6 .

Method 1100 includes steps 1102, 1104, 1106, and 1108, which may occurin any order. In step 1102, a first tooth, of the first frequency comb,is locked to the first laser at a first non-zero offset frequencysynthesized from a RF frequency reference. The first tooth may bephase-locked to the first laser by controlling a first CEO frequency ofthe first frequency comb. In one example of step 1102, LO tooth 406(1)is phase-locked to first reference laser 410(1) at the first frequencyoffset Δf_(A) ^((LO)) (see FIG. 4 ) by controlling the CEO frequency f₀^((LO)). In step 1104, a second tooth, of the first frequency comb, islocked to a second laser at a second non-zero offset frequencysynthesized from the RF frequency reference. The second tooth may bephase-locked to the second laser by controlling a first comb spacing ofthe first frequency comb. In one example of step 1104, LO tooth 406(2)is phase-locked to second reference laser 410(2) at the second frequencyoffset Δf_(B) ^((LO)) by controlling the repetition rate f_(rep)^((LO)).

In step 1106, a third tooth, of the second frequency comb, is locked tothe first laser at a third non-zero offset frequency synthesized fromthe RF frequency reference. The third tooth may be phase-locked to thefirst laser by controlling a second CEO frequency of the secondfrequency comb. In one example of step 1106, SIG tooth 404(1) isphase-locked to first reference laser 410(1) at the third frequencyoffset Δf_(A) ^((SIG)) by controlling the CEO frequency f₀ ^((SIG)). Instep 1108, a fourth tooth, of the second frequency comb, is locked tothe second laser at a fourth non-zero offset frequency synthesized fromthe RF frequency reference. The fourth tooth may be phase-locked to thesecond laser by controlling a second comb spacing of the secondfrequency comb. In one example of step 1108, SIG tooth 404(2) isphase-locked to second reference laser 410(2) at the fourth frequencyoffset Δf_(B) ^((SIG)) by controlling the repetition rate f_(rep)^((SIG)).

Method 1100 also includes a step 1110 in which the first, second, third,and fourth offset frequencies are selected so that a sum of the secondand third offset frequencies equals a sum of the first and fourth offsetfrequencies. In step 1110, the first, second, third, and fourth offsetfrequencies are also selected so that the first and second frequencycombs form a plurality of Nyquist windows. In each of the Nyquistwindows, the first frequency comb has an integer number of teeth equalto a frame length of interferograms recorded by the dual frequency-combspectrometer. In one example of step 1110, the offset frequencies Δf_(A)^((LO)), Δf_(B) ^((LO)), Δf_(A) ^((SIG)), and Δf_(B) ^((SIG)) of FIG. 4are selected to meet the no-walking condition of Eqn. 12, and to formlower and upper anchor frequencies that define Nyquist windows in whichthe number of LO teeth 406 is one greater than the number of SIG teeth404.

In one embodiment, method 1100 includes a step 1112 to synthesize thefirst, second, third, and fourth offset frequencies from the RFfrequency reference. Step 1110 and/or step 1112 may occur before orafter steps 1102, 1104, 1106, and 1108. In another embodiment, method1100 includes a step 1114 to spectroscopically detect a gas with thedual frequency comb spectrometer. In one example of step 1114, DCSsystem 100 uses DCS spectrometer 102, configured to meet the no-walkingcondition of Eqn. 12, to detect gas 116.

Interferogram Data Analysis

FIGS. 12-17 illustrate method embodiments for analyzing interferogramsthat facilitate streamlined data acquisition and real-time datafiltering and analysis. Advantageously, these method embodiments can beused to track the SNR as it changes over time, as may occur due tovarying weather conditions (e.g., wind, temperature, precipitation,etc.). Since changes in the SNR directly affect the sensitivity withwhich DCS 100 can detect gas 116, knowledge of how the SNR is changingallows parameters of DCS system 100 to be adjusted accordingly to ensurethat operation of DCS system 100 remains optimized at all times. Forexample, when interferograms have a low SNR, measurement time may beincreased to collect additional interferograms, as needed to reach atarget sensitivity. Alternatively, the target sensitivity may bedecreased if the required measurement time is too long. Similarly, whenthe interferograms have a high SNR, measurement time can be reducedsince fewer interferograms are needed to achieve the target sensitivity.Alternatively, the target sensitivity may be increased to, for example,determine a quantity of gas 116 with more sensitivity or to place a morestringent lower limit on the presence of gas 116. Methods illustrated inFIGS. 12-17 may be combined in any way without departing from the scopehereof. In addition, dual frequency-comb spectrometer 102 may beconfigured to implement any of these methods, either individually or inany combination.

FIG. 12 illustrates a method for analyzing an interferogram 1200 toquantify the SNR of interferogram 1200. Specifically, a signal amplitude1202 of interferogram 1200 may be obtained from the difference ofmaximum and minimum recorded values of a center burst 1206 ofinterferogram 900. A noise level 1204 may be obtained by calculating ormeasuring the root-mean-square (rms) variation in data points away fromcenter burst 1206. The SNR may be obtained by dividing noise level 1204into signal amplitude 1202.

FIG. 13 illustrates a method for using an upper threshold 1310 and alower threshold 1320 to identify and reject an interferogram 1300 with asignal amplitude too small to provide reliable data. An interferogramwith insufficient SNR may be identified when its center burst does notextend upward past upper threshold 1310 and/or downward below lowerthreshold 1320. In the example of FIG. 13 , a first center burst 1306(1)of a first interferogram 1300(1) extends neither upward past upperthreshold 1310 nor downward below lower threshold 1320, and thus firstinterferogram 1300(1) may be rejected. However, a second center burst1306(2) of a second interferogram 1300(2) extends both upward past upperthreshold 1310 and downward below lower threshold 1320, and thus secondinterferogram 1300(2) has a sufficient SNR to be retained.

While FIG. 13 shows upper threshold 1310 and lower threshold 1320 assymmetrically spaced about the zero-signal level, upper threshold 1310and lower threshold 1320 may alternatively be asymmetrically spacedabout the zero-signal level (i.e., an average of upper threshold 1310and lower threshold 1320 is non-zero). In one embodiment, only one ofupper threshold 1310 and lower threshold 1320 is used to rejectinterferograms 1300. In another embodiment, upper threshold 1310 and/orlower threshold 1320 are changed over time to account for changingsignal levels and/or SNR (e.g., due to changing environmentalconditions).

FIG. 14A illustrates a method for averaging a plurality ofinterferograms 1400 together to generate a SNR-enhanced interferogram1410 having a higher SNR than any one of interferograms 1400. Thus,SNR-enhanced interferogram 1410 can be used to determine a quantity ofgas 116 with a higher sensitivity than any one of interferograms 1400.Advantageously, SNR-enhanced interferogram 1410 contains less data thanall of interferograms 1400, and thus may be subsequently processedfaster than all interferograms 1400 without sacrificing SNR. Forexample, when each of interferograms 1400 contains N data points, all ofinterferograms 1400 contains N×M data points, where M is the number ofinterferograms 1400 included in the averaging. By contrast, SNR-enhancedinterferogram 1410 only contains N data points, which is less than N×Mby a factor of M. By improving processing speed, this data reductiontechnique enables real-time determination of the SNR.

Averaging interferograms 1400 reduces noise (i.e., noise level 1204 inFIG. 12 ) while maintaining signal (i.e., signal amplitude 1202). Togenerate SNR-enhanced interferogram 1410, each center burst 1406 shouldoccur at the same position in the corresponding interferogram 1400,otherwise the averaging will “wash out” center bursts 1406. To ensurethat center bursts 1406 always occur at the same position ininterferograms 1400, DCS spectrometer 102 can be operated under the“no-walking condition”, as described above (see Eqn. 12).

SNR-enhanced interferogram 1410 may be updated in real-time with newinterferograms 1400 as they are recorded. After a new interferogram 1400is averaged into SNR-enhanced interferogram 1410, it may be discarded.Furthermore, the SNR of SNR-enhanced interferogram 1410 may bedetermined by calculating a signal amplitude 1402 and a noise level 1404of SNR-enhanced interferogram 1410. In one embodiment, referred toherein as “adaptive SNR data acquisition”, averaging of newinterferograms 1400 into SNR-enhanced interferogram 1410 stops when theSNR of SNR-enhanced interferogram 1410 is equal to or greater than a SNRthreshold.

In an embodiment, DCS system 100 of FIG. 1 uses adaptive SNR dataacquisition to automate data collection along a plurality of paths(e.g., through or near gas 116). When the SNR of SNR-enhancedinterferogram 1410 equals or surpasses the SNR threshold, DCS system 100controls gimbal mount 110 to steer double pulse train 106 toward adifferent retroreflector 118, at which point SNR-enhanced interferogram1410 is reset and new interferograms 1400 are recorded and averaged toform a new SNR-enhanced interferogram 1410. This process may continueaccording to a preselected sequence of retroreflectors 118.

The SNR of SNR-enhanced interferogram 1410 increases with a number ofinterferograms 1106 included in the averaging when the noise of eachinterferogram 1406 includes a random component with zero mean. Noiselevel 1404 will continue to decrease with the number of interferograms1406 until the total acquisition time (equal to the product of thenumber of interferograms 1406 and the frame time T_(f)) approaches themutual coherence time of frequency combs 120(1), 120(2). When the mutualcoherence of frequency combs 120(1), 120(2) drifts, center bursts 1406may also drift, and the underlying phase of center bursts 1406 may slip.In either case, signal amplitude 1402 is reduced when averaginginterferograms 1410 obtained while the mutual coherence drifts.Similarly, if the short-term mutual coherence is low (i.e., the twocombs are relatively “noisy”), center bursts 1416 will “jump” from oneinterferogram 1400 to the next, and averaging of interferograms 1400will also reduce signal amplitude 1402.

FIG. 14B is a flow chart of a method 1450 for adaptive dualfrequency-comb spectroscopy. Method 1450 includes a step 1456 to recorda single interferogram with a dual frequency-comb spectrometer. In oneexample of step 1456, DCS spectrometer 120 detects retroreflected pulsetrain 108 to generate an interferogram (e.g., interferogram 302 of FIG.3 or any of interferograms 1400 of FIG. 14A). Method 1450 includes astep 1458 to average the single interferogram into an averagedinterferogram. In one example of step 1458, each of interferograms 1400is averaged into SNR-enhanced interferogram 1410. Method 1450 includes astep 1460 to determine a SNR of the averaged interferogram. In oneexample of step 1460, the SNR of the averaged interferogram isdetermined by (i) determining a signal amplitude of a center burst ofthe averaged interferogram (e.g., see signal amplitude 1402 of FIG.14A), and (ii) determining a noise level of the averaged interferogramfrom data points of the averaged interferogram located away from thecenter burst (e.g., see noise level 1404 of FIG. 14A). In anotherexample of step 1460, the SNR of the averaged interferogram isdetermined by (i) Fourier transforming the averaged interferogram into afrequency spectrum, and (ii) numerically integrating the frequencyspectrum (see FIG. 15 ). In some embodiments, method 1450 includes astep to discard the single interferogram after said averaging the singleinterferogram into the averaged interferogram; this step minimizesmemory usage by deleting the single interferogram after it is no longerneeded.

After step 1460, method 1450 continues with a decision 1462 thatcompares the SNR determined in step 1460 to a SNR threshold. The SNRthreshold is selected such that a trace gas can be detected with theaveraged interferogram to a desired sensitivity. If the SNR is less thanthe SNR threshold, then the SNR is not high enough to detect the tracegas at the desired sensitivity, and method 750 repeats steps 1456, 1458,and 1460 to record additional interferograms and increase the SNR of theaveraged interferogram with the additional interferograms.

In some embodiments, method 1450 includes a step 1454 to reset theaveraged interferogram prior to repeatedly recording, averaging, anddetermining (i.e., prior to a first iteration of steps 1456, 1458, and1460). The averaged interferogram may be reset by setting all datapoints of the averaged interferogram to zero. In some embodiments,method 1450 also includes a step 1452 to control a gimbal mount to steera double pulse train of the dual frequency-comb spectrometer to aretroreflector. In one example of 1452, gimbal mount 110 of DCS system100 is controlled such that double pulse train 106 retroreflects off ofretroreflector 118(1), as shown in FIG. 1 . While FIG. 14B shows step1454 occurring after 1452, step 1454 may alternatively occur afterbefore step 1452.

In some embodiments, decision 1462 determines that the SNR of theaveraged interferogram equals or exceeds the SNR threshold, whereinmethod 1450 continues with step 1452 to control the gimbal mount suchthat the double pulse train retroreflects off of a differentretroreflector (e.g., retroreflector 118(2) or retroreflector 118(3) ofFIG. 1 ). Thus, in these embodiments, operation of DCS system 100automatically updates to measure a new optical path through gas 116.

In some embodiments, method 1450 includes a step to operate the dualfrequency-comb spectrometer under a no-walking condition (see Eqn. 12above). By operating under this condition, the center bursts aresimilarly located within the recorded interferograms, and therefore donot cause the center-burst of the averaged interferogram to “wash-out”.

FIG. 15 illustrates how a frequency spectrum 1502 of an interferogram1500 may be alternatively used to determine the SNR of interferogram1500. A discrete Fourier transform 1510 (e.g., Fast Fourier transform,or FFT) may be applied to data points of interferogram 1500 to obtainFourier data points 1506 of frequency spectrum 1502. In FIG. 15 ,frequency spectrum 1502 only shows the amplitude, and not the phase, orFourier data points 1506, and thus Fourier data points 1206 arenon-negative. The SNR of interferogram 1500 may be obtained byintegrating frequency spectrum 1502, shown in FIG. 15 as a shaded area1504 under the curve formed by Fourier data points 1506. The SNRobtained from shaded area 1504 may be used similarly to the SNR obtainedby other methods, such as directly averaging interferograms 1400, asdescribed above for FIG. 14A.

FIG. 16 illustrates a method for gating an interferogram 1600 to reducea number of data points used for data processing and storage. Gated datapoints lying within a gate 1612 are retained for subsequent dataprocessing and storage, while ungated data points in portions 1604 lyingoutside of gate 1612 are discarded. Gate 1612 is centered near a centerburst 1606 such that the data points forming center burst 1606 areretained (i.e., are gated data points), while points away from centerburst 1606 (i.e., noise) are discarded.

Gating advantageously speeds up subsequent data processing by removingdata points that contain only, or predominantly, noise. The reduction ofdata points facilitates real-time data processing with a small,low-power processor (e.g., CPU, FPGA, SoC, etc.) collocated with DCS100. Thus, gating reduces the need for high-power offline computingresources needed for more intensive processing of larger data sets.Gating also advantageously reduces the amount of data needed to besubsequently stored, thereby reducing offline data storage requirements.

Gate 1612 is characterized by a temporal width and a temporal offsetwithin interferogram 1600. In one embodiment, both the temporal widthand the temporal offset are specified. After acquisition ofinterferogram 1600 is complete and the corresponding data points arestored in memory, gate 1612 may be applied to the stored data points toidentify and retain only those stored data points that are gated datapoints. These gated data points may then be forwarded for subsequentdata processing and storage. The ungated data points may be discardedfrom the memory, at which point the memory is ready to receive the nextinterferogram.

In one embodiment, only the temporal width of gate 1612 is specified.Data points from sequential interferograms are continuously stored in acircular buffer, and the presence of an interferogram is identified bydetecting a data point with a large signal value. For example, a firstdata point exceeding upper threshold 1310 (see FIG. 13 ) may signal thebeginning or leading edge of a center burst. The temporal offset of gate1612 may be selected based on the one data point to ensure that gate1612 captures all data points of the center burst.

FIG. 16 also shows an interferogram 1620 being gated with first andsecond gates 1632(1), 1632(2), thereby generating three disjointportions 1624 to be discarded. A second disjoint portion 1624(2)includes a center burst 1626 of interferogram 1620, and therefore firstand second gates 1632(1), 1632(2) are used to retain only the twodisjoint portions of interferogram 1620 adjacent to, but excluding,center burst 1626. Widths and offsets of gates 1632 may be selectedbased on the molecular species to be measured and/or desired reductionof data.

Gating with a plurality of gates 1632 may be advantageous when fittinginterferogram 1620 to a model in the time domain (as opposed to fittingthe Fourier transform of interferogram 1620 to a model in the frequencydomain). In this case, the gated data points have small values, andphotodetector gain may be increased to improve the SNR of the retaineddata points. With the higher gain, data points in second disjointportion 1624(2) may be allowed to saturate the photodetector (i.e.,photodetector 220 of FIG. 2 ) since these data points will be discarded.However, it is preferable that the photodetector recover quickly enoughsuch that data points in second gate 1632(2) are detected with the samelinearity as data points in first gate 1632(1).

FIG. 17 is a flow chart of a data-processing method 1700 for aninterferogram having a plurality of data points. Method 1700 may beimplemented with DCS system 100 of FIG. 1 . Method 1700 includes a step1702 to record the interferogram with a dual frequency combspectrometer. In one example of step 1702, photodetector 220 detectsdouble pulse train 106 to generate interferogram 302 of FIG. 3 . In oneembodiment, step 1702 includes a step 1704 to record the interferogramby spectroscopically detecting a gas with the dual frequency combspectrometer. In one example of step 1704, DCS system 100 of FIG. 1spectroscopically detects gas 116 by detecting double pulse train 106after double pulse train 106 has propagated through gas 116.

Method 1700 also includes a step 1706 to gate the interferogram. Step1706 includes a step 1708 to retain a plurality of gated data points ofthe data points of the interferogram, and a step 1710 to discard aremaining plurality of ungated data points. In one example of steps1706, 1708, and 1710, interferogram 1600 of FIG. 16 is gated with a gate1612, wherein data points of interferogram 1600 within gate 1612 areretained, and data points of interferogram 1600 lying in portions 1604outside of gate 1612 are discarded.

In one embodiment, method 1700 includes a step to detect a location of acenter burst of interferogram, wherein the interferogram is gated basedon the location of the center burst. In one example of this embodiment,a temporal width and a temporal offset of gate 1612 are chosen such thatcenter burst 1606 of interferogram 1600 lies within gate 1612, and thusdata points forming center burst 1606 are retained.

In other embodiments of method 1700, the interferogram is gated byretaining two disjoint sets of gated data points of the interferogram.In one example of these embodiments, interferogram 1620 of FIG. 16 isgated with disjoint first and second gates 1632(1), 1632(2). The twodisjoint sets are located on opposite sides of the center burst toexclude as least part of the center burst, such as shown withinterferogram 1620 of FIG. 16 . In one of these embodiments, the twodisjoint sets of gated data points are fitted to a time-domain model. Inanother of these embodiments, the method 1700 further includes selectinga gain of a photodetector such that the center burst saturates thephotodetector. These embodiments may be extended to retain three of moredisjoint sets of gated data points.

Improving Digitizer Nonlinearity by Shifting Interferograms

DCS system 100 operates over a wide range of weather conditions (e.g.,temperature, wind, precipitation, etc.) that create spatially- andtemporally-varying refractive indices that affect how pulse trains 106,108 propagate through free space. These variations focus, defocus,scatter, and/or distort pulse trains 106, 108, thereby causing some ofretroreflected pulse 108 train to miss optical transceiver 104 (andphotodetector 220). Thus, the SNR varies with weather conditions. It hasbeen observed by the inventors that repeatability of DCS system 100 isaffected by variations in the SNR. More specifically, two DCS systems100 measuring the same gas 116 under identical operating conditionsgenerate results (e.g., measured absorption levels) that do not agreewithin their statistical uncertainties, thereby signaling the presenceof a systematic source of error.

The inventors have recognized that to the above-mentioned systematicerror is caused by nonlinearity of an analog-to-digital converter (ADC)that digitizes the electrical signal outputted by photodetector 220 inresponse to detecting retroreflected double pulse 108. Accordingly,presented in this section are embodiments that improve the ADCnonlinearity, thereby advantageously reducing the observed systematicerror. This benefit has been confirmed experimentally by the inventors.Specifically, when the two DCS systems 100 are operated under identicalconditions and with the below embodiments implemented, agreement betweenthe results is significantly improved, thereby enhancing therepeatability of DCS system 100.

FIG. 18 shows a sine wave 1810 being added to a temporal sequence 1800to advantageously reduce integral nonlinearity of an ADC that digitizestemporal sequence 1800. Temporal sequence 1800 is a continuous analogsignal that corresponds, in the example of FIG. 18 , to teninterferograms 1802, each with a duration of T_(f). That is,interferograms 1802 are generated when temporal sequence 1800 isdigitized by the ADC. A sum of sine wave 1810 and temporal sequence 1800is shown in FIG. 18 as a shifted sequence 1812. A frequency of sine wave1810 is chosen such that an integer number of periods of sine wave 1810equals a duration of temporal sequence 1800. In the example of FIG. 18 ,where the duration of temporal sequence 1800 is 10T_(f), the frequencyof sine wave 1810 is 1/(10T_(f)).

In shifted sequence 1812, center bursts 1806 are shifted up and down,and therefore will be given different digital codes when digitized, ascompared to temporal sequence 1800. When interferograms 1802 are summed(e.g., when averaged, as described above in reference to FIG. 14A), thesignal shifts introduced by sine wave 1810 sum to zero. Thus, in theexample of FIG. 18 , the sum of interferograms 1802 of temporal sequence1800 nominally equals the sum of interferograms 1802 of shifted sequence1812.

However, in shifted sequence 1812, center bursts 1806 are digitizedacross a wider range of code values, as compared to temporal sequence1800, and thus more fully sample a transfer function of the ADC,including any nonlinear component of the transfer function. In some ADCarchitectures, the nonlinear component has odd symmetry about themid-scale code (corresponding to 0 signal in FIG. 18 ). In this case,when interferograms 1802 are summed, the nonlinear component of thetransfer function is reduced. Thus, the use of shifted sequence 1812advantageously improves linearity, as compared to temporal sequence1800.

FIGS. 19 and 20 show shifted sequences that are similar to shiftedsequence 1812 of FIG. 18 , except that sine wave 1810 has a higherfrequency. For the signal shifts introduced by sine wave 1810 to sum tozero, an integer number of periods of sine wave 1810 must equal aduration of temporal sequence 1800, as described above. While theexample of FIG. 18 shows the lowest frequency that meets this condition,the frequency may be alternatively set to a harmonic of the lowestfrequency. In the example of FIG. 19 , the frequency of sine wave 1810is 8/(10T_(f)), or the eighth harmonic of the lowest frequency. In theexample of FIG. 20 , the frequency of sine wave 1810 is 32/(10T_(f)), orthe thirty-second harmonic of the lowest frequency.

The examples of FIGS. 18-20 may be extended to an arbitrary number N ofconsecutive interferograms 1802 forming a shifted sequence.Specifically, the lowest frequency of sine wave 1810 is 1/(NT_(f)), andthe frequency of sine wave 1810 may be set to any integer harmonick/(NT_(f)) of the lowest frequency (i.e., k=1, 2, 3, . . . ). However,the frequency of sine wave 1810 may not equal the N^(th) harmonic, forwhich the frequency is N/(NT_(f))=11T_(f). In this case, where one cycleof sine wave 1810 corresponds to one of interferograms 1802, thedifferent cycles of sine wave 1810 will constructively interfere whenall interferograms 1802 are summed together, as opposed to summing tozero.

FIG. 21 is a functional diagram of an ADC nonlinearity canceler 2100that adds sine wave 1810 to temporal sequence 1802 to generate shiftedsequence 1812, digitizes shifted sequence 1812 into an interferogramsequence 2118, and averages interferograms 1802 of interferogramsequence 2118 to cancel signal shifts from sine wave 1810. Photodetector220 (see FIG. 2 ) outputs temporal sequence 1800 as an electrical signalwhen detecting retroreflected pulse train 108. Nonlinearity canceler2100 includes a frequency synthesizer 2106 that generates sine wave1810, and a signal combiner 2110 that combines temporal sequence 1800and sine wave 1810 to generate shifted sequence 1812. An ADC 2112 thendigitizes shifted sequence 1812, generating interferogram sequence 2118as digital data. Signal combiner 2110 may be a diplexer, a bias tee, asumming amplifier, or another type of circuit that sums two analogsignals.

ADC nonlinearity canceler 2100 also includes a signal processing circuit2114 that receives and processes digitized interferogram sequence 2118.Signal processing circuit 2114 includes a processor 2120 communicativelycoupled with a memory 2122 storing firmware 2130, interferogram sequence2118, and an averaged interferogram 2124. Firmware 2130 is implementedas machine-readable instructions that control processor 2120 to averageinterferograms 1802 stored in memory 2122 as interferogram sequence2118, and store the resulting average in memory 2122 as averagedinterferogram 2124. Firmware 2130 may also include instructions tooutput averaged interferogram 21, as shown in FIG. 21 , for subsequentprocessing (e.g., gating, phase correction, fitting, storage, etc.).

Processor 2120 may include a microprocessor chip, digital signalprocessor (DSP), field-programmable gate array (FPGA), or another typeof integrated circuit capable of performing logic, control, andinput/output operations. Memory 2122 may include both volatile memory(e.g., RAM, SRAM, etc.) and nonvolatile memory (e.g., ROM, FLASH, etc.).In some embodiments, signal processing circuit 2114 is implemented as amixed-signal integrated circuit, such as a system-on-chip (SoC) ormicrocontroller unit (MCU), that combines a processor, memory, andinput/output interfaces on a single chip. In some of these embodiments,the mixed-signal integrated circuit may include an ADC, wherein ADC 2112and signal processing circuit 2114 are implemented with the onemixed-signal integrated circuit.

ADC nonlinearity canceler 2100 includes a frequency controller 2108 thatcalculates the frequency of sine wave 2110 based on the frame time T_(f)and the number N of consecutive interferograms 1802 forminginterferogram sequence 2118. Frequency controller 2108 receives valuesfor N and T_(f) as parameters, and calculates therefrom the lowestfrequency of sine wave 1810. Alternatively, frequency controller 2108may receive values for f_(rep) ^((LO)), f_(rep) ^((SIG)), and/orΔf_(rep)=f_(rep) ^((LO))−f_(rep) ^((SIG)), and determine therefrom frametime T_(f) via T_(f)=1/Δf_(rep). Frequency controller 2108 alsodetermines if the frequency of sine wave 1810 should be set to thecalculated lowest frequency or a harmonic k thereof (except the N^(th)harmonic, as described above). As shown in FIG. 21 , frequencycontroller 2108 receives k as an input parameter. Alternatively,frequency controller 2108 may determine k based on the output frequencyrange of frequency synthesizer 2106 and/or any input bandwidthrequirements of signal combiner 2110. In one embodiment, frequencycontroller 2108 always selects the first harmonic k=1.

Frequency controller 2108 controls frequency synthesizer 2106 to outputsine wave 1810 according to the determined frequency. Frequencycontroller 2108 may control frequency synthesizer 2106 to continuouslyoutput sine wave 1810 until a new value for N, T_(f), and/or k isreceived, after which frequency controller 2108 may calculate an updatedfrequency of sine wave 1810, and control frequency synthesizer 2106 tooutput the updated frequency. In one embodiment, frequency controller2108 and signal processing circuit 2114 are implemented as one device.

ADC nonlinearity canceler 2100 may also include an amplitude controller2104 that interfaces with frequency synthesizer 2106 to set an amplitudeof sine wave 1810. As shown in FIG. 18 , shifted sequence 1812 shouldnever exceed an analog input range of ADC 2112. More specifically, amaximum signal 1814 of shifted sequence 1812 should not exceed a maximumanalog input of ADC 2112 (corresponding to a maximum code valueoutputted by ADC 2112). Similarly, a minimum signal 1816 of shiftedsequence 1812 should not fall below a minimum analog input of ADC 2112(corresponding to a minimum code value outputted by ADC 2112. In oneembodiment, amplitude controller 2104 sets the amplitude of sine wave1810 by driving a variable attenuator (not shown in FIG. 21 ) locatedbetween frequency synthesizer 2106 and signal combiner 2110. In anotherembodiment, amplitude controller 2104 and frequency controller 2108 areimplemented as one device. In another embodiment, amplitude controller2104, frequency controller 2108, and signal processing circuit 2114 areimplemented as one device.

FIG. 22 is a flow chart of a method 2200 for improving nonlinearity ofADC in a dual frequency-comb spectrometer. Method 2200 may beimplemented with ADC nonlinearity canceler 2100 of FIG. 21 . Method 2200includes a step 2208 to generate a shifted sequence by adding a sinewave to an analog temporal sequence recorded by a dual frequency-combspectrometer. In one example of step 2208, signal combiner 2110 of ADCnonlinearity canceler 2100 adds sine wave 1810 to temporal sequence 1800to generate shifted sequence 1812. Method 2200 also includes a step 2210to digitize the shifted sequence with an ADC to form a correspondinginterferogram sequence. In one example of step 2210, ADC 2112 digitizesshifted sequence 1812 to generate interferogram sequence 2218. Method2200 also includes a step 2212 to average the interferogram sequence toform an averaged interferogram free from shifts from the sine wave. Inone example of step 2212, signal processing circuit 2114 of ADCnonlinearity canceler 2100 averaged interferogram sequence 2118 to formaveraged interferogram 2124. In one embodiment, method 2200 includes astep 2214 to output the averaged interferogram. In one example of step2214, signal processing circuit 2114 outputs averaged interferogram2124, as shown in FIG. 21 .

In some embodiments, method 2200 includes a step 2204 to control afrequency of the sine wave such that an integer multiple of a period ofthe sine wave equals a duration of the interferogram sequence. In oneexample of step 2204, frequency controller 2108 of ADC nonlinearitycanceler 2100 interfaces with frequency synthesizer 2106 to control thefrequency of sine wave 1810. In one of these embodiments, the durationof the interferogram sequence is calculated from a number ofinterferograms forming the interferogram sequence and a frame time foreach of the interferograms.

In another embodiment, method 2200 includes a step 2206 to control anamplitude of the sine wave such that the shifted sequence does notexceed the analog input range of the ADC. In one example of step 2206,amplitude controller 2104 of ADC nonlinearity canceler 2100 interfaceswith frequency synthesizer 2106 to control the amplitude of sine wave1810.

In one embodiment, method 2200 includes a step 2202 to generate theanalog temporal sequence by spectroscopically detecting a gas with thedual frequency comb spectrometer. In one example of step 2202,photodetector 220 outputs temporal sequence 1800 as an electrical signalwhen detecting retroreflected pulse train 108.

Changes may be made in the above methods and systems without departingfrom the scope hereof. It should thus be noted that the matter containedin the above description or shown in the accompanying drawings should beinterpreted as illustrative and not in a limiting sense. The followingclaims are intended to cover all generic and specific features describedherein, as well as all statements of the scope of the present method andsystem, which, as a matter of language, might be said to falltherebetween.

What is claimed is:
 1. A data-processing method, comprising: recordingan interferogram with a dual frequency comb spectrometer, theinterferogram having a plurality of data points; and gating theinterferogram by: retaining a plurality of gated data points of the datapoints; and discarding a remaining plurality of ungated data points. 2.The data-processing method of claim 1, wherein said recording theinterferogram includes spectroscopically measuring a gaseous sample withthe dual frequency comb spectrometer.
 3. The data-processing method ofclaim 1, further comprising detecting a location of a center burst ofthe interferogram; wherein said gating the interferogram includes gatingthe interferogram based on the location of the center burst.
 4. Thedata-processing method of claim 3, wherein said gating the interferogramincludes retaining two disjoint sets of gated data points of theinterferogram.
 5. The data-processing method of claim 4, furthercomprising fitting the two disjoint sets of gated data points to atime-domain model.
 6. The data-processing method of claim 5, wherein thetwo disjoint sets are located on opposite sides of the center burst toexclude as least part of the center burst.
 7. The data-processing methodof claim 6, further comprising selecting a gain of a photodetector suchthat the center burst saturates the photodetector.
 8. A method foradaptive dual frequency-comb spectroscopy, comprising repeatedly:recording a single interferogram with a dual frequency-combspectrometer, averaging the single interferogram into an averagedinterferogram, and determining a signal-to-noise ratio (SNR) of theaveraged interferogram, until the SNR of the averaged interferogramexceeds a SNR threshold.
 9. The method of claim 8, said determining theSNR of the averaged interferogram includes: determining a signalamplitude of a center burst of the averaged interferogram; anddetermining a noise level of the averaged interferogram from data pointsof the averaged interferogram located away from the center burst. 10.The method of claim 8, said determining the SNR of the averagedinterferogram includes: Fourier transforming the averaged interferograminto a frequency spectrum; and numerically integrating the frequencyspectrum.
 11. The method of claim 8, further comprising discarding thesingle interferogram after said averaging the single interferogram intothe averaged interferogram.
 12. The method of claim 8, furthercomprising operating the dual frequency-comb spectrometer under ano-walking condition.
 13. The method of claim 8, further comprisingcontrolling a gimbal mount to steer a double pulse train of the dualfrequency-comb spectrometer to a retroreflector.
 14. The method of claim13, wherein said controlling occurs prior to said repeatedly recording,averaging, and determining.
 15. The method of claim 14, furthercomprising resetting the averaged interferogram prior to said repeatedlyrecording, averaging, and determining.